/4ZA*r7\ 


ASTRONOMY  DEPT. 


NEWCOMB'S 

MATHEMATICAL  COURSE. 


/.    SCHOOL    COURSE. 

Algebra  for  Schools. 

Key  to  Algebra  for   Schools. 

Plane  Geometry  and  Trigonometry,  with  Tables. 

The  Essentials  of  Trigonometry. 

II.    COLLEGE    COURSE. 

Algebra  for  Colleges. 

Key  to  Algebra  for  Colleges. 

Elements  of  Geometry. 

Plane  and  Spherical  Trigonometry,  with  Tables. 

Trigonometry   (separate). 

Tables  (separate). 

Elements  of  Analytic  Geometry. 

Elements  of  the  Differential  and  Integral  Calculus. 

Astronomy,  Advanced  Course,  by  Newcomb  and  Holden. 

Astronomy,  Briefer  Course,  by  Newcomb  and  Holden. 


HENRY  HOLT  &  CO.,  Publishers,  New  York. 


NE WC OMB' 8    MATHEMATICAL     COURSE 

LOGARITHMIC  AND  OTHER 
MATHEMATICAL    TABLES 


WITH  EXAMPLES  OF  THEIR  USE  AND  HINTS  ON  THE  ART  OK 
COMPUTATION 


BY 

SIMON    NEWCOMB 

f  Mathematics,  in  the  Johns  Hopkins  University. 


NEW  YORK 
HENRY  HOLT  AND  COMPANY 

LC  I  ?  I 


ASTRONOMY  DEPT 


COPYRIGHT,  1883, 

BY 
HENRY  HOLT  &  COl 


PBEFACE. 


IK  the  present  work  an  attempt  is  made  to  present  to  computers 
and  students  a  set  of  logarithmic  and  trigonometric  tables  which 
shall  have  all  the  conveniences  familiar  to  those  who  use  German 
tables.  The  five-figure  tables  of  F.  G.  GAUSS,  of  which  fifteen  edi- 
tions have  been  issued,  have,  after  long  experience  with  them,  been 
taken  as  the  basis  of  the  present  ones,  but  modifications  have  been 
introduced  wherever  any  improvement  could  be  made. 

Five  places  of  decimals  have  been  adopted  as  an  advantageous 
mean.  The  results  obtained  by  them,  being  nearly  always  reliable  to 
the  10,000th  part,  are  amply  accurate  for  most  computations,  while 
the  time  of  the  student  who  uses  them  is  not  wasted  in  unnecessary 
calculation. 

The  Introduction  is  intended  to  serve  not  only  as  an  explanation 
of  the  tables,  but  as  a  little  treatise  on  the  art  of  computation,  and 
the  methods  by  which  the  labor  of  computation  may  be  abridged. 

To  avoid  fostering  the  growing  evil  of  nearsightedness  among 
students,  the  author  and  publishers  have  spared  neither  pains  nor 
expense  in  securing  clearness  of  typography. 


M192073 


CONTENTS. 


INTRODUCTION  TO  TABLES. 

TABLE  I. 
LOGARITHMS  OP  NUMBERS. 

KAMI 

Introductory  Definitions 3 

The  Use  of  Logarithms 4 

Arrangement  of  the  Table  of  Logarithms 6 

Characteristics  of  Logarithms 8 

Interpolation  of  Logarithms 10 

Labor-Saving  Devices ' 11 

Number  Corresponding  to  Given  Logarithm 13 

Adjustment  of  Last  Decimal 14 

The  Arithmetical  Complement 16 

Practical  Hints  on  the  Art  of  Computation 18 

Imperfections  of  Logarithmic  Calculation 20 

Applications  to  Compound  Interest  and  Annuities 25 

Accumulation  of  an  Annuity 28 

TABLE  II. 

MATHEMATICAL  CONSTANTS. 
Explanation 31 

TABLES  III.  AND  IV. 
LOGARITHMS  OP  TRIGONOMETRIC  FUNCTIONS. 

Angles  less  than  45° 32 

Angles  between  45°  and  90° 33 

Angles  greater  than  90° 35 

Methods  of  Writing  the  Algebraic  Signs 36 

Angle  Corresponding  to  a  Given  Function 37 

Cases  when  the  Function  is  very  Small  or  Great 38 

TABLE  V. 

NATURAL  SINES  AND  COSINES. 
Explanation 42 


CONTENTS. 

TABLE  VI. 
ADDITION  AND  SUBTRACTION  LOGARITHMS. 

PAGE 

Use  in  Addition  ........................................................  43 

Use  in  Subtraction  ..................  t  ..................................     44 

Special  Cases  ...........................................................  45 


TABLE 
SQUARES  OP  NUMBERS. 
Explanation.  ..............  ..................  ..........................  49 

TABLE  VIII. 

HOURS,  MINUTES,  AND  SECONDS  INTO  DECIMALS  OP  A  DAY. 
Explanation  ...........................................  .  ................  51 

TABLE  IX. 

To  CONVERT  TIME  INTO  ARC. 
Explanation  ............................................................  53 

TABLE  X. 

MEAN  AND  SIDEREAL  TIME. 
Explanation  ............................................................  55 

OF  DIFFERENCES  AND  INTERPOLATION. 

General  Principles  ........  .  ......  .'  ......................................  56 

Fundamental  Formulae  ..................................................  61 

Transformations  of  the  Formulae  .........................................  62 

Formulas  of  Stirling  and  Bessel  .....................................  .....  68 

Special  Cases  of  Interpolation  —  Interpolation  to  Halves  ....................  64 

Interpolation  to  Thirds  ................................................  66 

Interpolation  to  Fifths  ...........................  ,  .....................  70 

FORMULAE  FOR  THE  SOLUTION  OF  PLANE  AND  SPHERICIAL 

TRIANGLES. 
Remarks  ...............................................................  74 

Formulae  ...............................................................  75 

TABLES  I.  TO  X. 


TABLE  I. 
LOGARITHMS   OF  NUMBERS. 


1.    Introductory  Definitions. 

Natural  numbers  are  numbers  used  to  represent  quantities. 

The  numbers  used  in  arithmetic  and  in  the  daily  transactions  of 
life  are  natural  numbers. 

To  every  natural  number  may  be  assigned  a  certain  other  number, 
called  its  logarithm. 

The  logarithm  of  a  natural  number  is  the  exponent  of  the 
power  to  which  some  assumed  number  must  be  raised  to  produce  the 
first  number.  The  assumed  number  is  called  the  base.  E.g.f  the 
logarithm  of  100  with  the  base  10  is  2,  because  102  =  100;  with  the 
base  2,  the  logarithm  of  64  would  be  6,  because  2fl  =  64. 

A  system  of  logarithms  means  the  logarithms  of  all  POSN 
tive  numbers  to  a  given  base. 

Although  there  may  be  any  number  of  systems  of  logarithms, 
only  two  are  used  in  practice,  namely: 

1.  The  natural  or  Napierian  system,  base  =  e  =  2. 718  282. 

2.  The  common  system,  base  =  10. 

The  natural  system  is  used  for  purely  algebraic  purposes. 

The  common  system  is  used  to  facilitate  numerical  calculations 
and  is  the  only  one  employed  in  this  book. 

If  the  natural  number  is  represented  by  n,  its  logarithm  is  called 

log  n. 

A  logarithm  usually  consists  of  an  integer  number  and  a  decimal 
part. 

The  integer  is  called  the  characteristic  of  the  logarithm. 

The  decimal  part  is  called  the  mantissa  of  the  logarithm. 

A  table  of  logarithms  is  a  table  by  which  the  logarithm  of 
any  given  number,  or  the  number  corresponding  to  any  given  loga- 
rithm, may  be  found. 


4  LOGARITHMIC  TABLES. 

The  moot  simple  form  of  table  is  that  on  the  first  page  of  Table  I.,  which 
gives  the  logarithms  of  all  entire  numbers  from  1  to  150;  each  logarithm  being 
found  alongside  its  number.  The  student  may  begin  his  exercises  with  this 
table. 

Mathematical  tables  in  general  enable  us,  when  one  of  two  related 
quantities  is  given,  to  find  the  other. 

In  such  tables  the  quantity  supposed  to  be  given  is  called  the 
argument. 

The  argument  is  usually  printed  on  the  top,  bottom,  or  side  of 
the  table. 

The  quantities  to  be  found  are  called  functions  of  the  argu- 
ment, and  are  found  in  the  same  columns  or  lines  as  the  argument, 
feut  in  the  body  of  the  table. 

In  a  table  of  logarithms  the  natural  number  is  the  argument, 
and  the  logarithm  is  the  function. 


2.     The  Use  of  Logarithms. 

The  following  properties  of  logarithms  are  demonstrated  in 
treatises  on  algebra. 

I.   The  logarithm  of  a  product  is  equal  to  the  sum  of  the  loga- 
rithms of  its  factors. 

II.   The  logarithm  of  a  quotient  is  found  by  subtracting  the  loga- 
rithm of  the  divisor  from  that  of  the  dividend. 

III.  The  logarithm  of  any  power  of  a  number  is  equal  to  the  loga- 
rithm of  the  number  multiplied  by  the  exponent  of  the  power. 

IV.  The  logarithm  of  the  root  of  a  number  is  equal  to  the  loga- 
rithm of  the  number  divided  by  the  index  of  the  root. 

"We  thus  derive  the  following  rules: 

To  find  the  product  of  several  factors  by  logarithms. 

EULE.  Add  the  logarithms  of  the  several  factors.  Enter  the 
table  with  the  sum  as  a  new  logarithm,  and  find  the  number  corres- 
ponding to  it. 

This  number  is  the  product  required. 

Example  1.     To  multiply  7x8. 

We  find  from  the  first  page  of  Table  I. 

log?  =  0.84510 

"   8  =  0.90309 


Sum  of  logs  =  1.748  19  =  log  of  product. 
Having  added  the  logarithms,  we  look  in  column  log  for  a  num- 


THE  USE  OF  LOGARITHMS.  5 

ber  corresponding  to  1.788  19  and  find  it  to  be  56,  which  is  the  pro- 
duct required. 

Ex.  2    To  find  the  continued  product  2x6x8. 
log  2,  0.30103 
"  6,  0.77815 
"  8,  0.90309 


Sum  of  logs,  1.982  27  =  log  product. 

The  number  corresponding  to  this  logarithm  is  found  to  be  96, 
which  is  the  product  required. 

Ex.  3.     To  find  the  quotient  of  147  -f-  21. 
log  147,  2.16732 
"     21,1.32222 

Difference,  0.845  10 

We  find  this  difference  to  be  the  logarithm  of  7,  which  is  tha 
required  quotient. 

Ex.  4.     To  find  the  quotient  arising  from  dividing  the  continued 
/roduct  98  X  102  X  148  by  the  continued  product  21  X  37  X  68. 
log  21,  1.322  22  log    98,  1.991  23 

"  37,  1.56820  "   102,  2.00860 

"  68,  1.83251  "   148,  2.17026 


Sum  =  log  divisor,  4.722  93  Sum  =  log  dividend,  6.170  09 

log  divisor,    4.72293 

Difference  =  log  quotient,  1.447  16 

Looking  into  the  table,  we  find  the  number  corresponding  to  this 
logarithm  to  be  28,  which  is  the  required  quotient. 

NOTE.  The  student  will  notice  that  we  have  found  this  quotient  without 
actually  determining  either  the  divisor  or  dividend,  having  used  only  their  loga- 
rithms. If  he  will  solve  the  problem  arithmetically,  he  will  see  how  much 
shorter  is  the  logarithmic  process. 

Ex.  5.     To  find  the  seventh  power  of  2. 
We  have  log  2  =  0.301  03 

7 

2.10721  =  log  128 
Hence  128  is  the  required  power. 
Ex.  6.     To  find  the  cube  root  of  125. 

3  |  2.09691 
0.69897 


6  LOGARITHMIC  TABLES. 

The  index  of  the  root  being  3,  we  divide  the  logarithm  of  125  by 
it.  Looking  in  the  tables,  we  find  the  number  to  be  5,  which  is  the 
root  required. 

EXEKCISES. 

Compute  the  following  products,  quotients,  powers,  and  roots  by 
logarithms. 

OO       C» 

1.  11 . 13.     Ans.  143.  5.  ~~.    Ans.  128. 


2.  12'.     Ans.  144.  6.  51 '  98^81.    Ans.  31. 

O4:  .    DO 

8.  -^-.     Ans.  48.  7.       '     .     Ans.  144. 

D  D 

2.  9'.  91.  78  ,    54.48 

4      13'.  21. 3    *     AnS'108-  8--8T9-'     Ans'36' 

3.  Arrangement  of  the  Table  of  Logarithms. 

A  table  giving  every  logarithm  alongside  its  number,  as  on  the 
first  page  of  Table  I.,  would  be  of  inconvenient  bulk.  For  numbers 
larger  than  150  the  succeeding  parts  of  Table  I.  are  therefore  used. 
Here  the  first  three  figures  of  the  natural  number  are  given  in  the 
left-hand  column  of  the  table.  The  first  figure  must  be  understood 
where  it  is  not  printed.  The  fourth  figure  is  to  be  sought  in  the 
horizontal  line  at  the  top  or  bottom.  The  mantissa  of  the  logarithm 
is  then  found  in  the  same  line  with  the  first  three  digits,  and  in  the 
column  having  the  fourth  digit  at  the  top. 

To  save  space  the  logarithm  is  not  given  in  the  column, 
but  only  its  last  three  figures.  The  first  two  figures  are  found  in 
the  first  column,  and  are  commonly  the  same  for  all  the  logarithms 
in  any  one  line. 

Example  1.  To  find  the  logarithm  of  2090. 

.  We  find  the  number  209,  the  figure  2  being  omitted  in  printing, 
in  the  left-hand  column  of  the  table,  and  look  in  the  column  having 
the  fourth  figure,  0,  at  its  top  or  bottom.  In  this  column  we  find 
320  15,  which  is  the  mantissa  of  the  logarithm  required. 

Ex.  2.    To  find  the  logarithm  of  2092. 

Entering  the  table  with  209  in  the  left-hand  column,  and  choos- 
ing the  column  with  2  at  the  top,  we  find  the  figures  056.  Te 
these  we  prefix  the  figures  32  in  column  0,  making  the  total  logarithm 
•320  56.  Therefore 

Mantissa  of  log  2092  =  .32056. 


ARRANGEMENT  OF  THE  TABLE.  7 

EXERCISES. 

Find  in  the  same  way  the  mantissas  of  the  logarithms  of  the  fol- 
lowing numbers: 

2240;  5133; 

2242;  5256; 

2249;  5504; 

2895;  8925; 

3644;  9557; 

4688;  9780. 

When  the  first  two  figures  of  the  mantissa  are  not  found  in  the 
same  line  in  which  the  number  is  sought,  they  are  to  be  found  in  the 
first  line  above  which  contains  them. 

Example.  The  first  two  figures  of  log  6250  are  79,  which  be- 
longs to  all  the  logarithms  below  as  far  as  6309.  Therefore  mantissa 
of  log  6250  =  .795  88. 

EXEECISES. 

Find  the  mantissae  of  the  logarithms  of 

6300;  answer,  .79934. 

6309;  "        .79996. 

6434; 

6653; 

6755; 

6918; 

7868. 

Exception.  There  are  some  cases  in  which  the  first  two  figures 
change  in  the  course  of  the  line.  In  this  case  the  first  two  figures 
are  to  be  sought  in  the  line  above  before  the  change  and  in  the  line 
next  below  after  the  change. 

Example.  The  mantissa  of  log  6760  is  .82995.  But  the  man- 
tissa of  log  6761  is  .83001.  In  this  case  the  figures  83  are  to  be 
found  in  the  next  line  below.  To  apprise  the  computer  of  these 
cases,  each  of  the  logarithms  in  which  the  two  first  figures  are  found 
in  the  line  below  is  indicated  by  an  asterisk. 

EXERCISES, 
Find  the  mantissa  of 

log  1022;     answer,  .009  45. 
log  1024;          "       .01030. 


8  LOGARITHMIC  TABLES. 

1231;  1999; 

1387;  3988; 

1419;  4675; 

1621;  4798; 

1622;  5377; 

1862;  8512; 

1863;  1009. 

4.    Characteristics  of  Logarithms. 

The  part  of  the  table  here  described  gives  only  the  mantissa  oj 
<each  logarithm.  The  characteristic  must  be  found  by  the  general 
theory  of  logarithms. 

The  following  propositions  are  explained  in  treatises  on  algebra: 
The  logarithm  of         1    is    0. 
"  "          "       10     "    1. 

"  "          "     100     "    2. 

"  "          "   1000     "    3. 

«          «          «       10*  "    n. 

Since  any  number  of  one  digit  is  between  0  and  10,  its  logarithm 
is  between  0  and  1;  that  is,  it  is  0  plus  some  fraction.  In  the  same 
way,  the  logarithm  of  a  number  of  two  digits  is  1  +  a  fraction.  And 
in  general, 

The  characteristic  of  the  logarithm  of  any  number  greater  than  1  is 

less  by  unity  than  the  number  of  its  digits  preceding  the  decimal  point. 

Example.    The  characteristic  of  the  logarithm  of  any  number 

between  1  and  10  is  0;  between  10  and  100  it  is  1;  between  100  and 

1000  it  is  2,  etc. 

Characteristic  of  log  1646         is    3. 
"  "    "      164.6      "    2. 

"  "    "        16.46     "    1. 

"  "    "          1.646  "    0. 

It  is  also  shown  in  algebra  that  if  a  number  be  divided  by  10  we 
diminish  its  logarithm  by  unity. 

Logarithms  of  numbers  less  than  unity  are  most  conveniently  ex- 
pressed by  making  the  characteristic  alone  negative. 
For  example: 

log  0.2    =  log  2  -  1  =  -  1  +  .301  03; 
"    0.02  =  log2-2  =  -2-f  .301  03. 

Hence:  The  mantissa  of  the  logarithms  of  all  numbers  which 
differ  only  in  the  position  of  the  decimal  point  are  the  same. 


CHARACTERISTICS  OF  LOGARITHMS.  9 

Hence,  also,  in  seeking  a  logarithm  from  the  table  we  find  the 
mantissa  without  any  reference  to  the  decimal  point.  Afterward  we 
affix  the  characteristic  according  to  the  position  of  the  decimal  point. 
For  convenience,  when  a  negative  characteristic  is  written  the 
minus  sign  is  put  above  it  to  indicate  that  it  extends  only  to  the 
characteristic  below  it  and  not  to  the  mantissa.  Thus  we  write 

log  .02  =  2. 301  03. 

In  practice,  however,  it  is  more  common  to  avoid  the  use  of 
negative  characteristics  by  increasing  them  by  10.  We  then  write 

log. 02  =  8. 301  03  -10. 

If  we  omitted  to  write  —  10  after  the  logarithm,  the  latter  would, 
in  strictness,  be  the  log  of  2  X  108.  But  numbers  so  great  as  this 
product  occur  so  rarely  in  practice  that  it  is  not  generally  neces- 
sary to  write  —  10  after  the  logarithm.  This  may  be  understood. 

A  convenient  rule  for  remembering  what  characteristic  belongs  to 
the  logarithm  of  a  decimal  fraction  is: 

The  characteristic  is  equal  to  9,  minus  the  number  of  zeros  after 
the  decimal  point  and  before  the  first  significant  figure. 
Examples,     log  34060  =4.53224 

"       340.60  =2.53224 

"  3.4060          =0.53224 

"  .03406       =8.53224-10 

"  .000  340  6  =  6.532  24  -  10 

It  will  be  seen  that  we  can  find  the  logarithms  of  numbers  from 
1  to  150  without  using  the  first  page  of  the  table  at  all,  since  all  the 
mantissas  on  this  page  are  found  on  the  following  pages  as  loga- 
rithms of  larger  numbers. 

EXERCISES. 

Find  the  logarithms  of  the  following  numbers: 
1.515  .003  899 

.01 702  0.4276 

18.62  464700 

.03  735  98.030 

Find  the  numbers  corresponding  to  the  following  logarithms: 


3.241  80; 
1.19145; 
A65321; 
€.74827; 
7.560  03; 

ans. 
ans. 
ans. 

450  000 
5  601  000 
36  310-000 

8.75035 
7.411  28 
.6.88997 
9.116  94 
7.250  18 

-10; 
-10; 
-.I?; 
-10; 

9.99991  - 
5.99996; 
2.96028; 
"0788627;" 
0.00087. 

10; 

10  LOGARITHMIC  TABLES. 


5.    Interpolation  of  Logarithms. 

In  all  that  precedes  we  have  used  only  logarithms  of  numbers 
containing  not  more  than  4  significant  digits.  But  in  practice 
numbers  of  more  than  four  figures  have  to  be  used.  To  find  the 
logarithms  of  such  numbers  the  process  of  interpolation  is  necessary. 
This  process  is  one  of  simple  proportion,  which  can  be  seen  from  the 
following  example. 

To  find  log.  1167.23. 

The  table  gives  the  logarithms  of  1167  and  of  1168,  which  we  find 
to  be  as  follows: 

log  1167  =  3.06707 

"  1168  =  3.06744 

Difference  of  logarithms  =    .000  37 

Now  the  number  of  which  we  wish  to  find  the  logarithm  being 
between  these  numbers,  its  logarithm  is  between  these  logarithms; 
that  is,  it  is  equal  to  3.067  07  plus  a  fraction  less  than  .000  37. 

Since  the  difference  37  corresponds  to  the  difference  of  unity  in 
the  two  numbers,  we  assume  that  the  quantity  to  be  added  to  the 
logarithm  bears  the  same  proportion  to  .23  that  37  does  to  unity. 
We  therefore  state  the  proportion 

1  :  .23  ::  37  :   increase  required. 

The  solution  of  this  proportion  gives  .23  X  37  =  8.51,  which  is 
the  quantity  to  be  added  to  log  1167  to  produce  the  logarithm 
required.*  The  result  is  3.067 155 1. 

But  our  logarithms  extend  only  to  five  places  of  decimals,  while 
the  result  we  have  written  has  seven.  "We  therefore  take  only  five 
places  of  decimals.  If  we  write  the  mantissa  3.06715,  the  result  will 
be  too  small  by  .51.  If  we  write  3.067  16,  it  will  be  too  great  by  .49. 
Since  the  last  result  is  nearer  than  the  first,  we  give  it  the  prefer, 
ence,  and  write  for  the  required  logarithm 

log  1167. 23  =  3.06716. 

We  thus  have  the  following  rule  for  interpolating: 

Take  from  the  table  the  logarithm  corresponding  to  the  first  four 
significant  digits  of  the  number. 

Considering  the  following  digits  as  a  decimal  fraction,  multiply 
the  difference  between  the  logarithm  and  the  next  one  following  by 
such  decimal  fraction. 

*  In  this  multiplication  we  have  used  a  decimal  point  to  mark  oil  the 
fifth  order  of  decimals.  This  is  a  convenient  process  in  all  such  computations. 


LABOR-SAVING  DEVICES.  H 

This  product  "being  added  to  the  logarithm  of  the  table  will  give 
the  logarithm  required. 

The  whole  operation  by  which  we  have  found  log  1167.23  would 
then  be  as  follows: 

log  1167  =  3.06707 
37  X  0.2  7.4 

X  0.03  1.11 


log  1167.23  =  3.06716 

The  products  for  interpolation,  7.4  and  1.11,  may  be  found  by 
multiplying  by  the  fifth  and  sixth  figures  of  the  number  separately. 

To  facilitate  this  multiplication,  tables  of  proportional  parts  are 
given  in  the  margin.  Each  difference  between  two  logarithms  will 
be  readily  found  in  heavy  type  not  far  from  that  part  of  the  table 
which  is  entered,  and  under  it  is  given  its  product  by  .1,  .2,  etc., . .  .9. 
We  therefore  enter  this  little  table  with  the  fifth  figure,  and  take  out 
the  corresponding  number  to  be  added  to  the  logarithm.  Then  if 
there  is  a  sixth  figure,  we  enter  with  that  also  and  move  the  decimal 
one  place  to  the  left.  We  then  add  the  two  sums  to  the  logarithm. 

6.    Labor-saving  Devices. 

In  using  a  table  of  logarithms,  the  student  should  accustom 
himself  to  certain  devices  by  which  the  work  may  be  greatly  facili- 
tated. 

In  the  first  place  it  is  not  necessary  to  take  the  whole  difference 
between  two  consecutive  logarithms.  He  has  only  to  subtract  the 
last  figure  of  the  preceding  logarithm  from  the  last  one  of  the  fol- 
lowing, increased  by  10  if  necessary,  and  thus  find  the  last  figure  of 
the  difference. 

The  nearest  difference  in  the  margin  of  the  table  having  this 
same  last  figure  will  always  be  the  difference  required. 

Example.  If  the  first  four  figures  of  the  number  are  1494,  in- 
stead of  subtracting  435  from  464  we  say  5  from  14  leaves  9,  and 
look  for  the  nearest  difference  which  has  9  for  its  last  figure.  This 
we  readily  find  to  be  29,  at  the  top  of  the  next  page. 

NOTE.  In  nearly  all  cases  the  difference  will  be  found  on  the  same  page 
with  the  logarithm.  The  only  exception  is  at  the  bottom  of  the  first  page,  wheret 
owing  to  the  number  of  differences,  they  cannot  all  be  printed. 

In  the  preceding  examples  we  have  written  down  the  numbers  in 
full,  which  it  is  well  that  the  beginner  should  do  for  himself.  But 
after  a  little  practice  it  will  be  unnecessary  to  write  down  anything 


12  LOGAEITHMIG  TABLES. 

but  the  logarithm  finally  taken  out.  The  student  should  accustom 
himself  to  take  the  proportional  parts  mentally,  adding  them  to  the 
logarithm  of  the  table  and  writing  down  the  sum  at  sight  The 
habit  of  doing  this  easily  and  correctly  can  be  readily  acquired  by 
practice. 

Exercises.    Find  the  logarithms  of 

792  638;  0.99997; 

1000.77;  949.916; 

1000.07;  20.8962; 

100  007;  660  652; 

181 982;  77.642; 

281.936;  8.8953. 

As  a  precaution  in  taking  out  logarithms,  the  computer  should 
always,  after  he  has  got  his  result,  look  into  the  table  and  see  that 
it  does  really  fall  between  two  consecutive  logarithms  in  the  table. 

If  the  fraction  to  be  interpolated  is  nearly  unity,  especially  if  it 
is  equal  to  or  greater  than  9,  it  will  generally  be  more  convenient  to 
multiply  the  difference  of  the  logarithms  by  the  complement*  of  the 
fraction  and  subtract  the  product  from  the  logarithm  next  succeed- 
ing. The  following  are  examples  of  the  two  methods,  which  may 
always  be  applied  whether  the  fraction  be  large  or  small: 
Example  1.  log  1004.28  =  log  (1005  -  .72). 

log  1004,         .00173  log  1005,         .00217 

pr.  pt.  for         .2,  8.8         pr.  pt.  for        .7,        -  30.8 

"    "    "  .08,  3.5          "    "    "          .02,       —      .9 

log,  3.001  85  log,  3.001  85 

Ex.  2.     log  154  993  =  155  000  -  7. 

log  1549,         .19005                             1550,         .19>033 
pr.  pt.  for         .9,              25.2       pr.  pt.  for     —  .07,         —  1.9b 
"     "    "  .03,  0.8  


log,  5.19031 


log,  5.19031 


*  By  the  complement  or  arithmetical  complement  of  a  decimal  fraction  is  here 
meant  the  remainder  found  by  subtracting  it  from  unity  or  from  a  unit  of  the 
next  order  higher  than  itself.  Thus : 

co.  .723    =  .277 

co.  .1796  =  .8204 

co.  .9932  =  .0068. 


NUMBER  CORRESPONDING  TO  A   GIVEN  LOGARITHM.       13 


7.    To  find  the  Number  corresponding  to  a  given 
Logarithm. 

The  reverse  process  of  finding  the  number  corresponding  to  a 
given  logarithm  will  be  seen  by  the  following  example: 

To  find  the  number  of  which  the  logarithm  is  2.027  90. 

Entering  the  table,  we  find  that  this  logarithm  does  not  exactly 
occur  in  the  table.  We  therefore  take  the  next  smaller  logarithm, 
which  we  find  to  be  as  follows: 

log  1066  =  2.02776. 

Subtracting  this  from  the  given  logarithm  we  find  the  latter  to  be 
greater  by  14,  while  the  difference  between  the  two  logarithms  of 
the  table  is  40.     We  therefore  state  the  proportion 
40  :  14  : :  1  to  the  required  fraction. 

The  result  is  obtained  by  dividing  14  by  40,  giving  a  quotient  .35. 
The  required  number  is  therefore  106.635.  It  will  be  remarked  that 
we  take  no  account  of  the  characteristic  and  position  of  the  decimal 
until  we  write  down  the  final  result,  when  we  place  the  decimal  in 
the  proper  position. 

The  table  of  proportional  parts  is  used  to  find  the  fifth  and  sixth 
figures  of  the  number  by  the  following  rule: 

If  the  given  logarithm  is  not  found  in  the  table,  note  the  ex- 
cess  of  the  given  logarithm  above  the  next  smaller  one  in  the  table, 
which  call  A. 

Take  the  difference  of  the  two  tabular  logarithms,  and  fjrui  it 
among  the  large  figures  which  head  the  proportional  parts. 

That  proportional  part  next  smaller  than  A  will  be  the  fifth 
figure  of  the  required  number. 

Take  the  excess  of  A  above  this  proportional  part;  imagine  its 
decimal  point  removed  one  place  to  the  right,  and  find  the  nearest 
number  of  the  table. 

This  number  will  be  the  sixth  figure  of  the  required  number. 

Example.    To  find  the  number  of  which  the  logarithm  is  2.193  59. 

Entering  the  table,  we  find  the  next  smaller  logarithm  to  be 
.193  40.  Therefore  A  =  19. 

Also  its  tabular  difference  =  28. 

Entering  the  table  of  proportional  parts  under  28,  we  find  16.8 
opposite  6  to  be  the  number  next  smaller  than  19  the  value  of  A. 
Therefore  the  fifth  figure  of  the  number  is  6. 

The  excess  of  19  above  16.8  is  2.2.  Looking  in  the  same  tablt 
for  the  number  22,  we  find  the  nearest  to  be  opposite  8. 


14  LOGARITHMIC  TABLES. 

Therefore  the  fifth  and  sixth  figures  of  the  required  number  are 
68.  Now  looking  at  the  log  .193  40  and  taking  the  corresponding 
number,  we  find  the  whole  required  number  to  be 

156  168. 

The  characteristic  being  2,  the  number  should  have  three  figures 
before  the  decimal  point.  Therefore  we  insert  the  decimal  point  at 
the  proper  place,  giving  as  the  final  result  156.168. 

8.    Number  of  Decimals  necessary. 

In  the  preceding  examples  we  have  shown  how  with  these  tables 
the  numbers  may  be  taken  out  to  six  figures.  In  reality,  however, 
it  will  seldom  be  worth  while  to  write  down  more  than  five  figures. 
That  is,  we  may  be  satisfied  by  adding  only  one  figure  to  the  four 
found  from  the  table.  In  this  case,  when  we  enter  the  table  of 
proportional  parts,  we  take  only  the  number  corresponding  to  the 
nearest  proportional  part. 

To  return  to  the  last  preceding  example,  where  we  find  the  num- 
be/  corresponding  to  2. 193  59.  We  find  under  the  difference  28  that 
th^  number  nearest  19  is  19.6,  which  is  opposite  7. 

Therefore  the  number  to  be  written  down  would  be  156.17. 

In  the  following  exercises  it  would  be  well  for  the  student  to 
4n  ite  six  figures  when  the  number  is  found  on  one  of  the  first  two 
pa^es  of  the  table  and  only  five  when  on  one  of  the  following  page* 
Tl  -e  reason  of  this  will  be  shown  subsequently. 

EXAMPLES  AND  EXERCISES. 

\.  To  find  the  square  root  of  f . 
We  have  log  3,  0.477  12 

"   2,  0.30103 

log  |,  0.17609 
-f-  2,  log  V$,  0.08804 

Here  we  have  a  case  in  which  the  half  of  an  odd  number  is 
required.  We  might  have  written  the  last  logarithm  0.088045,  but 
we  should  then  have  had  six  decimals,  whereas,  as  our  tables  only 
give  five  decimals,  we  drop  the  sixth.  If  we  write  4  for  the  fifth 
figure  it  will  be  too  small  by  half  a  unit,  and  if  we  write  5  it  will 
be  too  large  by  half  a  unit.  It  is  therefore  indifferent  which  figure 
we  write,  so  far  as  mere  accuracy  is  concerned. 


NUMBER  OF  DECIMALS  NECESSARY.  15 

A  good  rule  to  adopt  in  such  a  case  is  to  write  the  nearest  EVEN" 
number.  For  example, 

for  the  half  of  .261  81  we  write  .130  90; 
"  "  .26183  "  .13092; 
"  "  .26185  "  .13092; 
"  "  .26187  "  .13094; 
"  "  .26189  "  .13094; 
"  "  .26197  "  .13098; 
"  "  .26199  "  .13100. 

Returning  to  our  example,  we  find,  by  taking  the  number  corre- 
sponding to  0.088  04, 

Vt  =  1.224  72. 

2.  To  find  the  square  root  of  f . 

log  2,  0.301  03 
"  3,  0.47712 

logf,  9.82391  -  10 
|  log  |,  4.911  96  -  5  =  log  |/f. 
The  last  logarithm  is  the  same  as 

9.911  96  -  10, 

which  is  the  form  in  which  it  is  to  be  written  in  order  to  apply  the 
rule  of  characteristics.     The  corresponding  number  is  0.816  50. 

We  have  here  a  case  in  which,  had  we  neglected  considering  the 
surplus  —  10  as  we  habitually  do,  the  characteristic  of  the  answer 
would  have  been  4  instead  of  9  or  —  1.  The  easiest  way  to  treat 
such  cases  is  this: 

When  we  have  to  divide  a  logarithm  in  order  to  extract  a  root, 
instead  of  increasing  the  characteristic  by  10,  increase  it  by  10  X 
index  of  root. 

Thus  we  write        log  ^=  19.823  91  -  20. 
Dividing  by  2,  log  i/f  =    9.911  96  -  10,     • 

which  is  in  the  usual  form.  ,  > 

3.  To  find  the  cube  root  of  |. 

logl,  0.00000 
"    2,  0.30103 

log|,  9.69897  -10, 
which  we  write  in  the  form 

log  |  =  29.69897-30. 
Dividing  this  by  3, 

J  log  1  =  log  VT  =    9-899  66  -  10. 


16  LOGARITHMIC  TABLES. 

This  logarithm  is  in  the  usual  form,  and  gives 
V?=  0.793  70. 

The  affix  —  30,  or  —  10  x  divisor,  can  be  left  to  be  understood 
in  these  cases  as  in  others.  All  that  is  necessary  to  attend  to  is  that 
instead  of  supposing  the  characteristic  to  be  one  or  more  units  less 
than  10,  as  in  the  usual  run  of  cases,  we  suppose  it  to  be  one  or  more 
nnits  less  than  10  X  divisor. 

Find:  4.  The  square  root  of  -J; 

5.  The  cube  root  of      2; 

6.  The  fourth  root  of  f  ; 

7.  The  fifth  root  of    20; 

8.  The  tenth  root  of  10; 

9.  The  tenth  root  of      . 


9.    The  Arithmetical  Complement. 

When  a  logarithm  is  subtracted  from  zero,  the  remainder  is 
called  its  arithmetical  complement. 

If  L  be  any  logarithm,  its  arithmetical  complement  will  be  —  L. 

Hence  if 

L  =  log  n, 
then 

arith.  comp.  =  —  L  =  log  -; 

that  is, 

The  arithmetical  complement  of  a  given  logarithm  is  the  logarithm 
tfthe  reciprocal  of  the  number  corresponding  to  the  given  logarithm. 

Notation.  The  arithmetical  complement  of  a  logarithm  is  writ- 
ten co-log.  It  is  therefore  defined  by  the  form 

co-log  n  =  log  —  . 

Finding  the  arithmetical  complement.  To  find  the  arithmetical 
Complement  of  log  2  =  0.301  03,  we  may  proceed  thus: 

0.00000 
log  2,  0,30103 

co-log  2,  9.69897-10. 

We  subtract  from  zero  in  the  usual  way;  but  when  we  come  to 
the  characteristic,  we  subtract  it  from  10.  This  makes  the  re* 
mainder  too  large  by  10,  so  we  write  —  10  after  it,  thus  getting  a 
quantity  which  we  see  to  be  log  0.5. 

We  may  leave  the  —  10  to  be  understood,  as  already  explained. 


THE  ARITHMETICAL  COMPLEMENT.  17 

The  arithmetical  complement  may  be  formed  by  the  following 
rule: 

Subtract,  each  figure  of  the  logarithm  from  9,  except  the  last  sig- 
nificant one,  which  subtract  from  10.  The  remainders  will  form  the 
arithmetical  complement. 

For  example,  having,  as  above,  the  logarithm  0.301 03,  we  form, 
mentally,  9-0  =  9;  9-3  =  6;  9-0  =  9;  9-1  =  8;  9-0  =  9; 

10  —  3  =  7;  and  so  write 

9.698  97 

as  the  arithmetical  complement. 

To  form  the  arithmetical  complement  of  3.284  00  we  have  9  —  3 
=  6;  9  —  2  =  7;  9  —  8  =  1;  10  —  4  =  6.  The  complement  is 

therefore 

6.71600. 

The  computer  should  be  able  to  form  and  write  down  the  arith- 
metical complement  without  first  writing  the  tabular  logarithm,  the 
subtraction  of  each  figure  being  performed  mentally. 

Use  of  the  arithmetical  complement.  The  co-log  is  used  to  substi- 
tute addition  for  subtraction  in  certain  cases,  on  the  principle:  To 
add  the  co-logarithm  is  the  same  as  to  subtract  the  logarithm. 

Example.  We  may  form  the  logarithm  of  J  in  this  way  by  ad- 
dition: 

log  3,  0.47712 
co-log  2,  9.69897 

logj,  0.17609 

Here  there  is  really  no  advantage  in  using  the  co-log.  But  there 
is  an  advantage  in  the  following  example: 

97fi3   N/  J.1Q  9J. 

To  find  the  value  of  P  =  ;L          •     We  ad<*  to  the  loga- 

yy 

rithms  of  the  numerator  the  co-log  of  the  denominator,  thus: 
log  2763,       3.44138 
log    419.24,  2.62246 
co-log      99,       8.00436 

logP,  4.06820 

'.P  =  11700. 

The  use  of  the  arithmetical  complement  is  most  convenient  when 
^o  divisor  is  a  little  less  than  some  power  of  10. 


J8  LOQAEITHMIC  TABLES. 

EXERCISES. 

Form  by  arithmetical  complements  the  values  of: 

109  X  216.26 


1. 
2. 
3. 


0.99316 
8263  X  9162.7 

92  X  99.618 
4  X  6  X  8219 
9X992 


1C.  Practical  Hints  on  the  Art  of  Computation. 

The  student  who  desires  to  be  really  expert  in  computation 
should  learn  to  reduce  his  written  work  to  the  lowest  limit,  and  to 
perform  as  many  of  the  operations  as  possible  mentally.  We  have 
already  described  the  process  of  taking  a  logarithm  from  the  table 
without  written  computation,  and  now  present  some  exercises  which 
will  facilitate  this  process. 

1.  Adding  and  subtracting  from  left  to  right.  If  one  has  but 
two  numbers  to  add  it  will  be  found,  after  practice,  more  easj  and 
natural  to  write  the  sum  from  the  left  than  from  the  right.  The 
method  is  as  follows: 

In  adding  each  figure,  notice,  before  writing  the  sum,  whether 
the  sum  of  the  figures  following  is  less  or  greater  than  9,  or  equal 
to  it. 

If  the  sum  is  less  than  9,  write  down  the  sum  found,  or  its  last 
figure  without  change. 

If  greater  than  9,  increase  the  figure  by  1  before  writing  it  down. 

If  equal  to  9,  the  increase  should  be  made  or  not  made  accord- 
ing as  the  first  sum  following  which  differs  from  9  is  greater  or  less 
•than  9. 

If  the  first  sum  which  differs  from  9  exceeds  it,  not  only  must  we 
increase  the  number  by  1,  but  must  write  zeros  under  all  the  places 
where  the  9's  occur.  If  the  first  sun  different  from  9  is  less  than  9, 
write  down  the  9's  without  change. 

The  following  example  illustrates  the  process: 

7502768357858892837 
8239171645041102598 

15741940002899995435 
Here  7  and  8  are  15.      5  +  2  being  less    than  9,  we  write  15  without 
change.    3  +  0  being  less  than  9,  we  write  7  without  change.    9  +  2  being 
greater  than  9,  we  increase  the  sum  3  +  0  by  1  and  write  down  4.  7  +  1  being 


PRACTICAL  HINTS  ON  THE  ART  OF  COMPUTATION.      19 


less  than  9,  we  write  the  last  figure  of  9  -|-  2,  or  1,  without  change.  6  +  7  being 
greater  than  9,  we  increase  7  +  1  by  1  and  write  down  9.  Under  6  +  7  we 
write  down  3  or  4.  To  find  which,  8+  1  =  9;  3  +  6  =  9;  5+4=  9;  7  +  5  = 
12.  This  first  sum  which  is  different  from  9  being  greater  than  9,  we  write  4 
under  6  +  7,  and  O's  in  the  three  following  places  where  the  sums  are  9.  7+5 
=  12.  Since  8  +  0  <  9,  we  write  down  2.  Before  deciding  whether  to  put  8  or 
9  under  8  +  0,  we  add  5  +  4  =  9;  8  +  1  =  9;  8  +  1  =9;  9  +  0  =  9;  2  +  2  =  4 
This  being  less  than  9,  we  write  8  under  8  +  0,  and  9's  in  the  four  following 
places.  Since  5  +  8  =  13  >  9,  we  write  5  under  2  +  2.  Since  9+  3  =  12  >  9,  we 
write  4  under  5  +  8.  Since  8  +  7  =  15  >  9,  we  write  3  under  9  +  3.  Finally, 
under  8  +  7  we  write  5. 

This  process  cannot  be  advantageously  applied  when  more  than 
two  numbers  are  to  be  added. 

EXERCISES. 

Let  the  student  practise  adding  each  consecutive  pair  of  the  fol- 
lowing lines,  which  are  spaced  so  that  he  can  place  the  upper  margin 
of  a  sheet  of  paper  under  the  lines  he  is  adding  and  write  the  sum 
•upon  it. 

250917285316981208 
251235964692184368 
791615832316646891 
208532164379102909 
868588964342944825 
987654321012345674 

Subtracting.  We  subtract  each  figure  of  the  subtrahend  from 
the  corresponding  one  of  the  minuend  (the  latter  increased  by  10  if 
necessary),  as  in  arithmetic. 

Before  writing  down  the  difference,  we  note  whether  the  follow- 
ing figure  of  the  subtrahend  is  greater,  less,  or  equal  to  the  corre- 
'gponding  figure  of  the  minuend. 

If  greater,  we  diminish  the  remainder  by  1  and  write  it  down.* 

If  less,  we  write  the  remainder  without  change. 

If  equal,  we  note  whether  the  subtrahend  is  greater  or  less  than 
the  minuend  in  the  first  following  figure  in  which  they  differ. 

If  greater,  we  diminish  the  remainder  by  1,  as  before,  and  write 
9's  under  the  equal  figures. 

*  If  the  student  is  accustomed  to  carrying  1  to  the  figures  of  the  minuend 
when  he  has  increased  the  figure  of  his  subtrahend  by  10,  he  may  find  it  easier 
to  defer  each  subtraction  until  he  sees  whether  the  remainder  is  or  is  not  to  be 
diminished  by  1,  and,  in  the  latter  case,  to  increase  the  minuend  by  1  before 
subtracting. 


20  LOGARITHMIC  TABLES. 

If  less,  write  the  remainder  unchanged,  putting  O's  under  the 
equal  figures. 

Example. 

72293516214394 
24268518014198 

48024998200196 

Here  7  —  2  =5;  because  4  >  2,  we  write  4  12  —  4  =  8;  because  2  =  2 
and  6  <  9,  we  write  8;  and  write  0  in  the  following  place.  9—6  =  3;  be- 
cause 8  >  3,  we  write  2.  13  —  8  =  5;  5  =  5;  1  =  1;  8>6;  so  under  13  —  8  we 
write  4,  with  9's  in  the  two  next  places.  16  —  8  =  8;  because  0  <  2,  we  write 
8.  2  —  0  =  2;  1  =  1;  4  =  4;  1  <  3;  so  under  2  —  0  we  write  2,  followed  by  O's. 
3  —  1  =  2;  because  9  =  9,  8  >  4,  we  write  1,  with  9  in  the  next  place.  14  —  8  = 
6,  which  we  write  as  the  last  figure. 

EXERCISES. 

The  preceding  exercises  in  addition  will  serve  as  exercises  in  sub- 
traction by  subtracting  each  line  from  that  above  or  below  it.  The 
student  should  be  able  to  subtract  with  equal  facility  whether  the 
minuend  is  written  above  or  below  the  subtrahend. 

Mental  addition  and  subtraction.  When  an  expert  computer  has 
to  add  or  subtract  two  logarithms,  as  in  forming  a  product  or  quo- 
tient of  two  quantities,  he  does  not  necessarily  write  both  of  them, 
but  prefers  to  write  the  first  and,  taking  the  other  mentally,  add  (or 
subtract)  each  figure  in  order  from  left  to  right,  and  write  down  the 
sum  (or  difference).  He  thus  saves  the  time  spent  in  writing  one 
number,  and,  sometimes,  the  inconvenience  of  writing  it  where 
there  is  not  sufficient  room  for  it. 

This  process  of  inverted  addition  is  most  useful  in  adding  the 
proportional  part  in  taking  a  logarithm  from  the  table.  It  is  then 
absolutely  necessary  to  save  the  computer  the  trouble  of  copying 
both  logarithm  and  proportional  part. 

Expert  computers  can  add  seven-figure  logarithms  in  this  way 
without  trouble.  But  with  those  who  do  not  desire  to  become  ex- 
perts it  will  be  sufficient  to  learn  to  add  two  or  three  figures,  so  as 
to  be  able  to  take  a  five-figure  or  seven -figure  logarithm  from  the 
table  without  writing  anything  but  the  result. 

11.  Imperfections  of  Logarithmic  Calculations. 

Nearly  all  practical  computations  with  logarithms  are  affected 
by  certain  sources  of  error,  arising  from  the  omission  of  deci- 
mals. It  is  important  that  these  errors  should  be  understood  in 


IMPERFECTIONS  OF  LOGARITHMIC  CALCULATIONS.        21 

order  not  only  to  avoid  them  so  far  as  possible,  but  to  avoid  spend- 
ing labor  in  aiming  at  a  degree  of  accuracy  beyond  that  of  which  the 
numbers  admit. 

Mathematical  results  may  in  general  be  divided  into  two  classes: 
(1)  those  which  are  absolutely  exact,  and  (2)  those  which  are  only 
to  a  greater  or  less  degree  approximate. 

As  an  example  of  the  former  case,  we  have  all  operations  upon 
entire  numbers  which  involve  only  multiplication  and  division.  For 
example,  the  equations 

16s  =  256 
j^_16 
6a~  9 
are  absolutely  exact. 

But  if  we  express  the  fraction  |  as  a  decimal  fraction,  we  have 
^  =  .142857.  .,  etc.,  ad  infinitum. 

Hence  the  representation  of  \  as  a  decimal  fraction  can  never  be 
absolutely  exact.  The  amount  of  the  error  will  depend  upon  how 
many  decimals  we  include.  If  we  use  only  two  decimals  we  shall 
certainly  be  within  one  hundredth;  if  three,  within  one  thou- 
sandth, etc.  Hence  the  degree  of  accuracy  to  which  we  attain  de- 
pends upon  the  number  of  decimals  employed.  By  increasing  the 
number  of  decimals  we  can  attain  to  any  degree  of  accuracy.  As  an 
example,  it  is  shown  in  geometry  that  if  the  ratio  of  the  circumfer- 
ence of  a  circle  to  its  diameter  be  written  to  35  places  of  decimals, 
the  result  will  give  the  whole  circumference  of  the  visible  universe 
without  an  error  as  great  as  the  minutest  length  visible  in  the  most 
powerful  microscope. 

There  are  no  numbers,  except  the  entire  powers  of  10,  of  which 
the  logarithms  can  be  exactly  expressed  in  decimals.  We  must 
therefore  omit  all  figures  of  the  decimal  beyond  a  certain  limit.  The 
number  of  decimals  to  be  used  in  any  case  depends  upon  the  degree 
of  accuracy  which  is  required.  The  large  tables  of  logarithms  con- 
tain seven  decimal  places,  and  therefore  give  results  correct  to  the 
ten-millionth  part  of  the  unit.  This  is  sufficiently  near  the  truth 
in  nearly  all  the  applications  of  logarithms. 

With  five  places  of  decimals  our  numbers  will  be  correct  to  the 
hundred-thousandth  part  of  a  unit.  This  is  sufficiently  near  for 
most  practical  applications. 

Accumulation  of  errors.  When  a  long  computation  is  to  be 
made,  the  small  errors  are  liable  to  accumulate  so  that  we  cannot 
rely  upon  this  degree  of  accuracy  in  the  final  result.  The  manner 


•22  LOGARITHMIC  TABLES. 

in  which  the  tables  are  arranged  so  as  to  reduce  the  error  to  a  mini- 
mum may  be  shown  as  follows: 

We  have  to  seven  places  of  decimals 

log  17  =  1.230  448  9 
"    18  =  1.2552725 

When  the  tables  give  only  five  places  of  decimals  the  two  last 
figures  must  be  omitted.  If  the  tables  gave  log  17=. 230  44,  the 
logarithm  would  be  too  small  by  89  units  in  the  seventh  place.  It  is 
therefore  increased  by  a  unit  in  the  fifth  place,  and  given  .23045. 
This  quantity  is  then  too  large  by  11,  and  is  therefore  nearer  the 
truth  than  the  other.  The  nearest  number  being  always  given,  we 
have  the  result: 

Every  logarithm  in  the  table  differs  from  the  truth  ~by  not  more 
•than  one  half  a  unit  of  the  last  place  of  decimals. 

Since  the  error  may  range  anywhere  from  zero  to  half  a  unit,  and 
is  as  likely  to  have  one  value  as  another  between  those  limits,  we 
-conclude: 

The  average  error  of  the  logarithms  in  the  tables  is  one  fourth  of 
<a  unit  of  the  last  place  of  decimals. 

Errors  in  interpolation.  When  we  interpolate  the  logarithm  we 
add  to  the  tabular  logarithm  another  quantity,  the  proportional  part, 
which  may  also  be  in  error  by  half  a  unit,  but  of  which  the  average 
•error  will  only  be  one  fourth  of  a  unit. 

As  most  logarithms  have  to  be  interpolated,  the  general  result 
will  be: 

An  interpolated  logarithm  may  possibly  be  in  error  by  a  unit  in 
the  last  place  of  decimals. 

The  sum  of  the  average  errors  will,  however,  be  only  half  a  unit. 
But  these  errors  may  cancel  each  other,  one  being  too  large  and  the 
other  too  small.  The  theory  of  probabilities  shows  that,  in  conse- 
quence of  this  probable  cancellation  of  errors,  the  average  error  only 
increases  as  the  square  root  of  the  number  of  erroneous  units  added. 

The  square  root  of  2  is  1.41. 

If,  therefore,  we  add  two  quantities  each  affected  with  a  probable 
error,  ±  .25,  the  result  will  be,  for  the  probable  error  of  the  sum, 

1.41  X  .25  =  0.35. 
We  therefore  conclude: 

The  average  error  of  a  logarithm  derived  from  the  table  by  inter- 
polation is  0.35  of  a  unit  of  the  last  place. 

Applying  the  above  rule  of  the  square  root  to  the  case  in  which 


IMPERFECTIONS  OF  LOGARITHMIC  CALCULATIONS.       23 

several  logarithms  are  added  or  subtracted  to  form  a  quotient,  we? 
find  the  results  of  the  following  table: 


No.  of  logs  added  or  subtracted. 

Average  error. 

1  

0.35 

2             

0.50 

3  

0.63 

4  

0.72 

5  

0.81 

6  

0.88 

7  

0.95 

8  

1.02 

9  

1.08 

10.. 

,   1.14 

From  this  table  we  see  that  if  we  form  the  continued  product  of 
eight  factors,  by  adding  their  logarithms  the  average  error  of  th& 
sum  of  the  logarithms  will  be  more  than  a  unit  in  the  last  place. 

As  an  example  of  the  accumulation  of  errors,  let  us  form  the* 
product  11  .  13. 

We  have  from  the  table 

log  11  =  1.041  39 
"  13  =  1.11394 

•log  product,  2.155  33 

We  see  that  this  is  less  than  the  given  logarithm  of  the  product 
143  by  a  unit  of  the  fifth  order.  But  if  we  use  seven  decimals  we, 
have  log  11,  1.0413927 

"  13,  1.1139434 

2.1553361 

Comparing  this  with  the  computation  to  five  places,  we  see  the 
source  of  the  error. 

If  the  numbers  with  which  we  enter  the  tables  are  affected  by 
errors,  these  errors  will  of  course  increase  the  possible  errors  of  the 
logarithms. 

In  determining  to  what  degree  of  accuracy  to  carry  our  results, 
we  have  the  following  practical  rule  : 

It  is  never  worth  while  to  carry  our  decimals  beyond  the  limit  of 
precision  given  ~by  the  tables,  which  limit  may  be  a  considerable  frac- 
tion of  the  unit  in  the  last  figure  of  the  tables. 

Let  us  have  a  logarithm  to  five  places  of  decimals,  1.92949,  of 
which  we  require  the  corresponding  number.  Entering  the  table,  wa 


24  LOGARITHMIC  TABLES. 

perceive  that  the  corresponding  number  is  between  85. 01  an4  85.0& 
If  this  logarithm  is  the  result  of  adding  a  number  of  logarithms, 
each  of  which  may  be  in  error  in  the  way  pointed  out,  we  may  sup- 
pose it  probably  affected  by  an  error  of  half  a  unit  in  the  last  figure 
and  possibly  by  an  error  of  a  whole  unit  or  more.  That  is,  its  true 
value  may  be  anywhere  between  92  948  and  92  950. 

The  number  corresponding  to  the  former  value  is  85.  C13,  and 
that  corresponding  to  the  latter  85.016.  Since  the  numbtrs  may 
fall  anywhere  between  these  limits,  we  assign  to  it  a  mean  \alue  of 
85. 014,  which  value,  however,  may  be  in  error  by  two  units  in  the 
last  place.  It  is  not,  therefore,  worth  while  to  carry  the  interpolation 
further  and  to  write  more  than  five  digits. 

Next  suppose  the  logarithm  to  be  2.021  70.  Entering  the  table, 
we  find  in  the  same  way  that  the  number  probably  lies  between  the 
limits  105.121  and  105.126.  There  is  therefore  an  uncertainty  of 
five  units  in  the  sixth  place,  or  half  a  unit  in  the  fifth  place.  If  the 
greatest  precision  is  desired,  we  should  write  105. 124.  But  our  last 
figure  being  doubtful  by  two  or  three  units,  the  question  might  arise 
whether  it  were  worth  while  to  write  it  at  all.  As  a  general  rule,  if 
the  sixth  figure  is  required  to  be  exact,  we  must  use  a  six-  or  seven- 
place  table  of  logarithms. 

Still,  near  the  beginning  of  the  table,  the  probable  error  will  be 
diminished  by  writing  the  sixth  figure. 

Now  knowing  that  at  the  beginning  of  the  table  a  difference  of 
one  unit  in  the  number  makes  a  change  ten  times  as  great  in  the 
logarithm  as  at  the  end  of  the  table,  we  reach  the  conclusions  : 

In  talcing  out  a  number  in  the  first  part  of  the  table,  it  can  never 
be  worth  while  to  write  more  than  six  significant  figures^  and  very 
little  is  added  to  the  precision  by  writing  more  than  five. 

In  the  latter  part  of  the  table  it  is  never  worth  while  to  write  more 
than  five  significant  figures. 

Sometimes  no  greater  accuracy  is  required  than  can  be  gained  by 
irj/ng  four-figure  logarithms.  There  is  then  no  need  of  writing  the 
last  figure.  If,  however  the  printed  logarithm  is  used  without 
change,  the  fourth  figure  must  be  increased  by  unity  whenever  the 
fifth  figure  exceeds  5.  When  the  fifth  figure  is  exactly  5,  the  increase 
should  or  should  not  be  made  according  as  the  5  is  too  small  or  too 
great.  To  show  how  the  case  should  be  decided,  a  stroke  is  printed 
above  the  5  when  it  is  too  great.  In  these  cases  the  fourth  figure 
should  be  used  as  it  stands,  but,  when  there  is  no  stroke,  it  should 
be  increased  by  unity. 


THE  COMPUTATION  OF  ANNUITIES,  ETC.  25 


12.  Applications  of  Logarithms  to  the  Computation  of 
Annuities  and  Accumulations  of  Funds  at  Com- 
pound Interest. 

One  of  the  most  useful  applications  of  logarithms  is  to  fiscal 
calculations,  in  which  the  value  of  moneys  accumulating  for  long 
periods  at  compound  interest  is  required. 

Compound  interest  is  gained  by  any  fund  on  which  the  interest 
is  collected  at  stated  intervals  and  put  out  at  interest. 

As  an  example,  suppose  that  $10  000  is  put  out  at  6  per  cent 
interest,  and  the  interest  collected  semi-annually  and  put  out  at  the 
same  rate.  The  principal  will  then  grow  as  follows: 

Principal  at  starting 810  000.00 

Six  months'  interest  =  3  per  cent 300.00 

Amount  at  end  of  6  months $10  300.00 

Interest  on  this  amount  =  3  per  cent. .         309.00 

Amount  at  end  of  1  year $10  609.00 

Interest  on  this  amount  =  3  per  cent. .         318.27 

Amount  at  end  of  H  years $10  927.27 

Interest  on  this  amount  for  6  months. .         327.82 


Amount  at  end  of  2  years $11  255.09 

Although  in  business  practice  interest  is  commonly  payable  semi- 
annually,  it  is  in  calculations  of  this  kind  commonly  supposed  to  be 
collected  and  re-invested  only  at  the  end  of  each  year.  This  makes 
the  computation  more  simple,  and  gives  results  nearer  to  those  ob- 
tained in  practice,  because  a  company  cannot  generally  invest  its 
income  immediately.  If  it  had  to  wait  three  months  to  invest  each 
semi-annual  instalment  of  interest  collected,  the  general  result  would 
be  about  the  same  as  if  it  collected  interest  only  once  a  year  and  in- 
vested it  immediately. 

If  r  be  the  rate  per  cent  per  annum,  the  annual  rate  of  increase 

will  be  — — .     Let  us  put 

1UU 

p,  the  annual  rate  of  increase  =  -r^; 

1UU 

p,  the  amount  at  interest  at  the  beginning  of  the  time,  or  the 
principal; 

a,  the  amount  at  the  end  of  one  or  more  years. 


26  LOGARITHMIC  TABLES. 

Then,  at  the  beginning  of  first  year,  principal p 

Interest  during  the  year pp 

Amount  at  end  of  year p  (1  -f-  p) 

Interest  on  this  amount  during  second  year pp  (I  -f-  p) 

Amount  at  end  of  second  year,  (1  +  p)p  (1  +  P)  —  P  (1  +  PY 
Continuing  the  process,  we  see  that  at  the  end  of  n  years  the 

amount  will  be 

a=p(l  +  py.  (1) 

To  compute  by  logarithms,  let  us  take  the  logarithms  of  both 
members.     We  then  have 

log  a  =  logp  +  n  log  (1  +  p).  (2} 

Example.   Find  the  amount  of  $1250  for  30  years  at  6  per  cent 
per  annum. 

Here  p  =    .06 

1  +  p  =  1.06 

log  (1  -f  p)  =  0.025  306  (end  of  Table  I.) 
30 


nlog(l+p),      0.75918 
Iogj9,      3.09691 

log  a,      3.85609 

a,  $7179.50  =  required  amount. 

EXERCISES. 

1.  Find  the  amount  of  $100  for  100  years  at  5  per  cent  compound 
interest. 

2.  A  man  bequeathed  the  sum  of  $500  to  accumulate  at  4  per 
cent  interest  for  80  years  after  his  death.  After  that  time  the  annual 
interest  was  to  be  applied  to  the  support  of  a  student  in  Harvard 
College.     What  would  be  the  income  from  the  scholarship? 

3.  If  the  sum  of  one  cent  had  been  put  out  at  3  per  cent  per 
annum  at  the  Christian  era,  and  accumulated  until  the  year  1800, 
what  would  then  have  been  the  amount,  and  the  annual  interest  on 
this  amount? 

It  is  only  requisite  to  give  three  significant  figures,  followed  by  the  necessary 
number  of  zeros. 

4.  Solve  by  logarithms  the  problem  of  the  horseshoeing,  in  which 
a  man  agrees  to  pay  1  cent  for  the  first  nail,  2  for  the  second,  and  so 
on,  doubling  the  amount  for  every  nail  for  32  nails  in  all. 

NOTE.  It  is  only  necessary  to  compute  the  amount  for  the  32d  nail,  be- 
cause it  is  easy  to  see  that  the  amount  paid  for  each  nail  is  1  cent  more  than  fnr 
all  the  preceding  ones. 


COMPUTATION  OF  ANNUITIES,  ETC.  .  27 

5.  A  man  lays  aside  $1000  as  a  marriage-portion  for  his  new-born 
daughter,  and  invests  it  so  as  to  accumulate  at  8  per  cent  compound 
interest.    The  daughter  is  married  at  the  age  of  25.    What  does  the 
portion  amount  to? 

6.  A  man  of  30  pays  $2000  in  full  for  a  $5000  policy  of  insurance 
on  his  life.    Dying  at  the  age  of  80,  his  heirs  receive  $7000,  policy 
and  dividends.     If  the  money  was  worth  4  per  cent  to  him,  how 
much  have  the  heirs  gained  or  lost  by  the  investment? 

7.  What  would  have  been  the  answer  to  the  previous  question, 
had  the  man  died  at  the  age  of  40,  and  the  amount  paid  been 
$6000? 

Other  applications  of  the  formula.  By  means  of  the  equations 
(1)  and  (2)  we  may  obtain  any  one  of  the  four  quantities  a,  p,  p,  and 
n  when  the  other  three  are  given. 

CASE  I.     Given  the  principal,  rate  of  interest,  and  time,  to 
find  the  amount. 

This  case  is  that  just  solved. 

CASE  II.    Given  the  amount,  time,  and  rate  per  cent,  to  find 
the  principal. 

Solution.    Equation  (1)  gives 


Taking  the  logarithms, 

log  p  =  log  a  —  n  log  (1  +  p), 
by  which  the  computation  may  be  made. 

CASE  III.    Given  the  principal,  amount,  and  time,  to  find  the  rate. 

Solution.     Equation  (2)  gives 


n  n        p 

Example.     A  man  wants  a  principal  of  $600  to  amount  to  $1000 
in  10  years.     At  what  rate  of  interest  must  he  invest  it? 
Solution.  log  a  =  3.000  00 

logp  =  2.77815 

log  -  =  0.221  85 

~  log|-  =  0.022  185  =  log  (1  +  P). 

Hence,  from  last  page  of  logarithms, 

1+  p  =  1.05241; 

and  rate  =  5.241, 

or  5^  per  cent,  nearly. 


28  LOGARITHMIC  TABLES. 

EXEKCISES. 

1.  At  what  rate  of  interest  will  money  double  itself  every  ten 
years?  Ans.  7.177. 

2.  At  what  rate  will  it  treble  itself  every  15  years?    Ans.  7.599. 

3.  A  man  having  invested  $1000,  with  all  the  interest  it  yielded 
him,  for  25  years,  finds  that  it  amounts  to  $3386.     What  was  the 
rate  of  interest?  Ans.  5  per  cent. 

4.  A  life  company  issued  to  a  man  of  20  a  paid-up  policy  for 
$10,000,  the  single  premium  charged  being  $3150.     If  he  dies  at  the 
age  of  60,  at  what  rate  must  the  company  invest  its  money  to  make 
itself  good?  Ans.  2.93  per  cent. 

5.  A  man  who  can  gain  4  per  cent  interest  wants  to  invest  such  a 
sum  that  it  shall  amount  to  $5000  when  his  daughter,  now  5  years  old, 
attains  the  age  of  20.     How  much  must  he  invest?    Ans.  $2776.  62. 

6.  How  much  must  a  man  leave  in  order  that  it  may  amount  to 
$1,000,000  in  500  years  at  2J  per  cent  interest?  Ans.  $4.36£ 

7.  How  much  if  the  time  is  1000  years,  the  rate  being  still  2-J 
per  cent,  and  the  amount  $1,000,000?  Ans.  0.0019  of  a  cent. 

8.  A  man  finds  that  his  investment  has  increased  fivefold  in  25 
years.     What  is  the  average  rate  of  interest  he  has  gained  ? 

Ans.  6.65. 

9.  An  endowment  of  $7500  is  payable  to  a  man  when  he  attains 
the  age  of  65.     What  is  its  value  when  he  is  45,  supposing  the  rate 
of  interest  to  be  4  per  cent?  Ans.  $3423. 

13.  Accumulation  of  an  Annuity. 

It  is  often  necessary  to  ascertain  the  present  or  future  value  of  a 
series  of  equal  annual  payments.  Thus  it  is  very  common  to  pay  a 
constant  annual  premium  for  a  policy  of  life  insurance.  The  value 
of  such  a  series  of  payments  at  any  epoch  is  found  by  reducing  the 
value  of  each  one  to  the  epoch,  allowing  for  interest,  and  taking  the 
sum.  Supposing  the  epoch  to  be  the  present  time,  the  problem  may 
be  stated  as  follows: 

A  man  agrees  to  pay  p  dollars  a  year  for  n  years,  the  first  pay* 
ment  being  due  in  one  year,  and  the  total  number  of  payments  n. 
What  is  the  present  value  of  all  n  payments  9 


Putting,  as  before,  p  =  rate  the  present  value  of  p 


dollars  payable  after  y  years  will,  by  §  12,  Case  II.,  be 

P 


ACCUMULATION  OF  AN  ANNUITY.  29 

Putting  in  succession,  y  =  1,  y  =  2,  .  .  .  y  =  w,  the  sum  of  the 
present  values  is 

p  p         ,          p         ,  ,    m      p 

1  4-  p  "•"  (i  -}-  p)a   '    (1  -f  p)3  "*  *"  (1  -f  p)»* 

This  is  a  geometrical  progression  in  which 

First  term  =  — ^- — ; 
1  +/> 

Common  ratio  =  ^         ; 

Number  of  terms  =  n. 
By  College  Algebra,  §  212,  the  sum  of  this  progression  will  be 

/      1       \n 

,      '-(iT-J 


(1  +  <»)•-! 


If  the  first  payment  is  to  be  made  immediately,  instead  of  at  the 
end  of  a  year,  the  last  or  «th  payment  will  be  due  in  n  —  1  years, 
and  the  progression  will  be 

p  +  r+p+  (i  +  />)*  +  '  '  *  +  (i+P)'1-1* 

We  find  the  sum  of  the  geometric  progression  to  be 


EXERCISES. 

1.  What  is  the  present  value  of  15  annual  payments  of  $85  each, 
of  which  the  first  is  due  in  one  year,  the  rate  being  5  per  cent? 
We  find  by  substitution 

Present  value  =  85  __!05^-1__      J5_     1.05»-  1 

1.0516-  1.0516      1.05"  '       .05 
_1700(1.0516-1) 
(1.05)  16        ' 

log  1.05,     0.021189  1.05",  2.07895 

_  15  1.05"  -1,    1.07895 

log  1.05",    0.31784  log,  0.03300 

co-log  1.0515,  9.68216 

log  1700,  3.23045 

Value,  $882.28  log  value,  2L945  6? 


30  LOQAR1THM10    TABLES. 

2.  The  same  thing  being  supposed,  what  would  be  the  present 
value  if  the  rate  of  interest  were  4  per  cent  ?  Ans.  $945.80 

3.  What  is  the  present  value  of  25  annual  payments  of  $1000 
each,  the  first  due  immediately,  if  the  rate  of  interest  is  3  per  cent  ? 

Ans.  $17,935 

4.  A  debtor  owing  $10,000  wishes  to  pay  it  in  10  equal  annual 
instalments,  the  first  being  payable  immediately.     If  the  rate  of 
interest  is  6  per  cent,  how  much  should  each  payment  be? 

Ans.  $1281.76. 

NOTE.  This  problem  is  the  reverse  of  the  given  one,  since,  in  the  equation 
(2),  we  have  given  2a  =  10000,  p  =  0.06,  and  n  =  10,  to  find  p. 

5.  The  same  thing  being  supposed,  what  should  be  the  annual 
payment  in  case  the  payments  should  begin  in  a  year? 

Ans.  $1358.69. 

Perpetual  annuities.  If  the  rate  of  interest  were  zero,  the 
present  value  of  an  infinity  of  future  payments  would  be  infinite. 
But  with  any  rate  of  interest,  however  small,  it  will  be  finite.  For  if, 

in  the  first  equation  (1),  we  suppose  n  infinite,  f  —  —  J  will  converge 
toward  zero,  and  we  shall  have 


This  result  admits  of  being  put  into  a  concise  form,  thus: 

Since  2  is  the  present  value  of  the   perpetual  annuity  p,  the 

annual  interest  on  this  value  will  be  p*S.     But  the  equation  (3)  gives 

p2=p. 

Hence: 

The  present  value  of  a  perpetual  annuity  is  the  sum  of  which  the 

annuity  is  the  annual  interest. 

Example.     If  the  rate  of  interest  were  3|  per  cent,  the  present 

value  of  a  perpetual  annuity  of  $70  would  be  $2000. 

EXERCISES. 

1.  A  government  owing  a  perpetual  annuity  of  $1000  wishes  to 
pay  it  off  by  10  equal  annual  payments.     If  the  rate  of  interest  is  4 
per  cent,  what  should  be  the  amount  of  each  payment? 

Ans.  $3082.30. 

2.  A  government  bond  of  $100  is  due  in  15  years  with  interest  at 
6  per  cent.     The  market  rate  of  interest  having  meanwhile  fallen  to 
3|  per  cent,  what  should  be  the  value  of  the  bond? 

NOTE.    We  find,  separately,  the  present  value  of  the  15  annual  instalments 
of  interest,  and  of  the  principal. 


TABLE  II. 
MATHEMATICAL  CONSTANTS. 


14.  In  this  table  is  given  a  collection  of  constant  quantities 
which  frequently  occur  in  computation,  with  their  logarithms. 

The  logarithms  are  given  to  more  than  five  decimals,  in  order  to 
be  useful  when  greater  accuracy  is  required.  When  used  in  five- 
place  computations,  the  figures  following  the  fifth  decimal  are  to  be 
dropped,  and  the  fifth  decimal  is  to  be  increased  by  unity  in  case 
the  figure  next  following  is  5  or  any  greater  one. 


TABLES  III.  AND  IT. 
LOGARITHMS  OF  TRIGONOMETRIC  FUNCTIONS. 


15.  By  means  of  these  tables  the  logarithms  of  the  six  trigono- 
metric functions  of  any  angle  may  be  found. 

The  logarithm  of  the  function  instead  of  the  function  itself  is 
given,  because  the  latter  is  nearly  always  used  as  a  factor. 

We  begin  by  explaining  Table  IV.,  because  Table  III.  is  used  only 
in  some  special  cases  where  Table  IV.  is  not  convenient. 

I.  Angles  less  than  45°.  If  the  angle  of  which  a  function  is 
soiight  is  less  than  45°,  we  seek  the  number  of  degrees  at  the  top  of 
the  table  and  the  minutes  in  the  left-hand  column.  Then  in  the 
line  opposite  these  minutes  we  find  successively  the  sine,  the  tan- 
gent, the  cotangent,  and  the  cosine  of  the  angle,  as  given  at  the 
heading  of  the  page. 

Example.  log  sin  31°  27'  =  9.717  47; 

log  tan  31°  27'  =  9.78647; 

log  cotan  31°  27'  =  0.213  53; 

cos  31°  27'  =  9.93100. 

The  sine,  tangent,  and  cosine  of  this  angle  being  all  less  than 
unity,  the  true  mantissas  of  the  logarithm  are  negative;  they  are 
therefore  increased  by  10,  on  the  system  already  explained. 

If  the  secant  or  cosecant  of  an  angle  is  required,  it  can  be  found 
by  taking  the  arithmetical  complement  of  the  cosine  or  sine.  It  is 
shown  in  trigonometry  that 

secant  =  — : — , 
cosine 

and 

cosecant  =  -; — . 
sine 

Therefore    log  secant  =  0  —  log  cosine  =  co-log  cosine; 

log  cosec  =  0  —  log  sine     =  co-log  sine. 
We  thus  find  log  sec  31°  27'  =  0.069  00; 

log  cosec  31°  27'  =  0.282  53. 
After  each  column,  upon  intermediate  lines,  is  given  the  differ- 


LOGARITHMS  OF  TRIGONOMETRIC  FUNCTIONS.  33 

ence  between  every  two  consecutive  logarithms,  in  order  to  facilitate 
interpolation. 

In  the  case  of  tangent  and  cotangent,  only  one  column  of  differ- 
ences is  necessary  for  both  functions. 

If  we  use  no  fractional  parts  of  minutes,  no  interpolation  is 
necessary;  but  if  decimals  of  a  minute  are  employed,  we  can  inter- 
polate precisely  as  in  taking  out  the  logarithms  of  numbers. 

"Where  the  differences  are  very  small  they  are  sometimes  omitted. 

Tables  of  proportional  parts  are  given  in  the  margin,  the  use  of 
which  is  similar  to  those  given  with  the  logarithms  of  numbers. 

Example  1.     To  find  the  log  sin  of  31°  27'.7. 

We  have  from  the  tables,  log  sin  31°  27'     =  9.717  47 
Under  diff.  20,  P.P.  for  7,  14 

log  sin  31°  27'.  7  =  9.71761 
Ex.  2.     To  find  log  cot  15°  44'. 34. 

The  tables  give     log  cot  15°  44'  =  0.550  19 
Under  diff.  48,  opposite  0.3,  P.P.,        -  14.4 
"  "         0.4  -7-  10,        -    1.9 

log  cot  15°  44'.  34,          0.55003 

Since  the  tabular  quantity  diminishes  as  the  angle  increases,  the 
proportional  parts  are  subtractive. 

EXEECISES. 
Find  from  the  tables : 

1.  log  cot  43°  29'. 3; 

2.  log  tan  43°  29'. 3; 

3.  log  cos  27°  10'.  6; 

4.  log  sin  27°  10'.  6; 

5.  log  tan  12°    9'. 43; 

6.  log  cot  12°    9'. 43. 

In  the  case  of  sines  and  tangents  of  small  angles  the  differences 
vary  so  rapidly  that  in  most  cases  the  exact  difference  will  not  be 
found  in  the  table  of  proportional  parts.  In  this  case,  if  the  pro- 
portional parts  are  made  use  of,  a  double  interpolation  will  generally 
be  necessary  to  find  the  fraction  of  a  minute  corresponding  to  a  given 
sine  or  tangent.  If  only  tenths  of  minutes  are  used,  an  expert  com- 
puter will  find  it  as  easy  to  multiply  or  divide  mentally  as  to  refer  to 
ihe  table. 

II.  Angles  between  45°  and  90°.  It  is  shown  in  trigonometry 
that  if  we  compute  the  values  of  the  trigonometric  functions  for  the 


34  LOGARITHMIC  TABLES. 

first  45°,  we  have  those  for  the  whole  circle  by  properly  exchanging 
them  in  the  different  parts  of  the  circle.    First,  if  we  have 

a  +  ft  =  90, 

then  a  and  ft  are  complementary  functions,  and 
|.  sin  ft  =  cos  a\ 

tan  ft  =  cotan  a. 

Therefore  if  our  angle  is  between  45°  and  90°,  we  may  find  its 
complement.  Entering  the  table  with  this  complement,  the  com- 
plementary function  will  then  be  the  required  function  of  the  angle. 

Example.  To  find  the  sine  of  67°  23',  we  may  enter  the  table 
with  22°  37'  (=  90°-  67°  23')  and  take  out  the  cosine  of  22°  37', 
whichfis  the  required  sine  of  67°  23. 

To  save  the  trouble  of  doing  this,  the  complementary  angles  and 
the  complementary  denominations  of  the  functions  are  given  at  the 
bottom  of  the  page. 

The  minutes  corresponding  to  the  degrees  at  the  bottom  are  given 
on  the  right  hand.  Therefore: 

To  find  the  trigonometric  functions  corresponding  to  an  angle 
between  45°  an d  90°,  we  take  the  degrees  at  the  bottom,  of  the  page  and 
the  minutes  in  the  right-hand  column.  The  values  of  the  four  func- 
tions log  sine,  log  tangent,  log  cotangent,  and  log  cosine,  as  read  at 
the  bottom  of  the  page,  are  then  found  in  the  same  line  as  the 
minutes. 

Example  1.     For  52°  59'  we  find 

log  sin  =  9.90225; 
log  tan  =  0.12262; 
log  cot  =  9.877  38; 
log  cos  =  9.77963. 
Ex.  2.  To  find  the  trigonometric  functions  of  77°  17'.  28. 


77°  17'        9. 

sin.                      tan.                     cot.                     cos. 

98921       0.64653       9.35347       9.34268 

P.P.  for  0.2 

+  0.6 

+  11.8 

-  11.8 

-  11.2 

"     0.08 

+  0.2 

+    4.7 

-    4.7 

-    4.5 

9. 989  22       0. 646  70       9. 353  30       9. 342  52 
Then  log  sec      =  co-log  cos  =  0.657  48; 

log  cosec  =  co-log  sin  =  0.010  78. 

EXERCISES. 
Find  the  logarithms  of  the  six  functions  of  the  following  angles: 

1.  45°  50'.  74;  3.     74°    0'.68; 

2.  48°49'.37;  4.     83°  59'.62. 


LOGARITHMS  OF  TRIGONOMETRIC 


III.     When  the  angle  exceeds  90°. 

EULE.  Subtract  from  the  angle  the  greatest  multiple  of  90° 
which  it  contains. 

If  this  multiple  is  180°,  enter  the  table  with  the  excess  of  the  angle 
over  180°  and  take  out  the  functions  required,  as  if  this  excess  were 
itself  the  angle. 

If  the  multiple  is  90°  or  270°,  take  out  the  complementary  func- 
tion to  that  required. 

By  then  assigning  the  proper  algebraic  sign,  as  shown  in  trigo- 
nometry, the  complete  values  of  the  function  will  be  obtained. 

The  computer  should  be  able  to  assign  the  proper  algebraic 
sign  according  to  the  quadrant,  without  burdening  his  memory  with 
the  special  rules  necessary  in  each  + 

case.     This  he  can  do  by  carrying  Sine  positive 

in  his  mind's  eye  the  following 
scheme.  He  should  have  at  com- 
mand the  arrangement  of  the  four 
quadrants  as  usually  represented  in 
trigonometry,  so  as  to  know,  when 
an  angle  is  stated,  where  it  will  fall 
relatively  to  the  horizontal  and  ver- 
tical lines  through  the  centre  of  the 
circle.  Then,  in  the  case  of — 

Sine  or  cosecant.     If  the  angle  Sine  negative 

is  above  the  horizontal  line  (which 
it  is  between  0°  and  ISO0),  the  sine  is  positive;  if  below,  negative. 

Cosine  or  secant.  If  the  angle  is  to  the  right  of  the  vertical 
central  line  (as  it  is  in  the  first  and  fourth  quadrants),  the  cosine  and 
secant  are  positive;  if  to  the  left  (as  in  the  second  and  third  quad- 
rants), negative. 

Tangent  or  cotangent.  Through  the  opposite  first  and  third  quad- 
rants, positive;  through  the  opposite  second  and  fourth  quadrants, 
negative. 

Example  1.  To  find  the  tangent  and  cosine  of  122°  44'.  Sub- 
tracting 90°,  we  enter  the  table  with  32°  44'  and  find 

log  cot  32°  44'  =  0.19192; 
log  sin  32°  44'  =  9.73298. 

T nerefore,  writing  the  algebraic  sign  before  the  logarithm,  we  hare 
log  tan  122°  44'  =  -  0.191  92; 
log  cos  122°  44'  =  —  9.732  98. 


86  LOGARITHMIC  TABLES. 

Ex.  2.    To  find  the  sine  and  cotangent  of  322°  58'. 
Entering  the  table  with  52°  58'  =  322°  58'  —  270°,  and  taking 
out  the  complementary  functions,  we  find 

log  sin  322°  58'  =  -  9.779  80; 
log  cot  322°  58'  =  —  0.122  36. 
Ex.  3.     To  find  the  sine  and  tangent  of  253°  5'. 
Entering  with  73°  5',  we  take  out  the  sine  and  tangent,  finding 
log  sin  253°  5'  =  —  9.890  79; 
log  tan  253°  5'  =  -f  0.516  93. 

Ex.  4.     To  find  the  six  trigonometric  functions  of  152°  38'.    We 
have 

log  sin  152°  38'  =  log  cos  62°  38'  pos.  =  +  9.662  46; 

log  cos  152°  38'  =  log  sin  62°  38'  neg.  =  —  9.948  45; 

log  tan  152°  38'  =  log  cot  62°  38'  neg.  =  —  9.71401; 

log  cot  152°  38'  =  log  tan  62°  38'  neg.  =  —  0.285  99; 

log  sec      =  co-log  cos  =  —  0.05155; 

log  cosec  =  co-log  sin  =  +  0.337  54. 

EXERCISES. 

Find  the  six  trigonometric  functions  of  the  following  angles: 

276°  29'.3; 

66°  0'.5; 

96°  59'.8; 

252°  20'.3; 

318°  10'.  7; 

-  25°  22'.2; 

-155°  30'.  7. 


16.  Method  of  Writing  the  Algebraic  Signs. 

As  logarithms  are  used  in  computation,  they  may  always  be  con- 
sidered positive.  It  is  true  that  the  logarithms  of  numbers  less  than 
unity  are  in  reality  negative,  but,  for  convenience  in  calculation,  we 
increase  them  by  10,  so  as  to  make  them  positive. 

The  number  corresponding  to  a  given  logarithm  may,  in  compu- 
tation, be  positive  or  negative.  There  are  two  ways  of  distinguishing 
the  algebraic  sign  of  the  number,  between  which  the  computer  may 
choose  for  himself. 

I.  Write  the  algebraic  sign  of  the  number  before  the  logarithm. 
As  usually  interpreted,  the  algebraic  sign  written  thus  would  apply 
to  the  logarithm,  which  it  does  not.  It  is  therefore  necessary  for  the 


ANGLE  CORRESPONDING   TO  A   GIVEN  FUNCTION.         37 

computer  to  bear  in  mind  that  the  sign  belongs,  not  to  the  loga- 
rithm, as  written,  but  to  the  number. 

II.  Write  the  letter  n  after  the  logarithm  when  the  number  i& 
negative.  This  plan  is  theoretically  the  best,  but,  should  the  com- 
puter accidentally  omit  the  letter,  the  number  will  be  treated  as 
positive,  and  a  mistake  will  be  made.  It  therefore  requires  vigilance 
on  his  part.  An  improvement  would  be  to  write  a  letter  not  likely 
to  be  mistaken  for  n,  s  for  instance,  after  all  positive  logarithms. 


17.  To  Find  the  Angle  Corresponding  to  a  Given 
Trigonometric  Function. 

Disregarding  algebraic  signs,  there  will  always  be  four  angles 
corresponding  to  each  function,  one  in  each  quadrant.  These  angles, 
will  be: 

The  smallest  angle,  as  found  in  the  table; 
This  angle  increased  by  180°; 
The  complementary  angle  increased  by  90°; 
The  complementary  angle  increased  by  270°. 

For  instance,  for  the  angle  of  which  log  tan  is  0.611  92,  we  find 
76°  16'.  But  we  should  get  this  same  tangent  for  103°  44',  256°  16', 
and  283°  44'. 

Of  the  four  functions  corresponding  to  the  four  angles,  two  will 
always  be  positive  and  two  negative;  so  that,  in  reality,  there  will 
only  be  two  angles  corresponding  to  a  function  of  which  both  the- 
sign  and  the  absolute  value  are  given.  These  values  are  found  by 
selecting  from  the  four  possible  ones  the  two  for  which  the  functions 
have  the  given  algebraic  sign.  After  selecting  them,  they  may  be 
checked  by  the  following  theorems,  which  are  easily  deduced  from: 
the  relations  between  the  values  of  each  function  as  given  in  trigo- 
nometry: 

The  sum  of  the  two  angles  corresponding  to  the  same  sine  is  ISO0' 
vr  540°. 

The  sum  of  the  two  angles  corresponding  to  the  same  cosine  is 
360°. 

The  difference  of  the  two  angles  corresponding  to  the  same  tangenk 
is  180°. 

Which  of  the  two  possible  angles  is  to  be  chosen  depends  upon 
the  conditions  of  the  problem  or  the  nature  of  the  figure  to  which 
the  angle  belongs.  If  neither  the  conditions  nor  the  figure  decida 
the  question,  the  problem  is  essentially  ambiguous,  and  either  ^~ 
Doth  angles  are  to  be  taken. 


38  LOGARITHMIC  TABLES. 


EXEECISES. 

Find  tne  pairs  of  values  of  the  angle  a  from  the  following  values 
of  the  trigonometric  functions: 

1.  log  sin  a  =  +  9.902  43;  12.  log  sec  a  =  -{-  0.221  06; 

2.  log  sin  a  =  -  9.902  43;  13.  log  sec  a  =  -  0.221  06; 

3.  log  cos  a  =  +  9.902  43;  14.  log  sec  a  -  -  0.099  20; 

4.  log  cos  a  =  —  9.902  43;  15.  log  sec  a  —  +  0.123  46; 

5.  log  tan  a  =  +  0.143  16;  16.  log  sin  a  =  -f  8.990  30; 

6.  log  tan  a  =  —  0.143  16;  17.  log  sin  a  ~  —  8.990  30; 

7.  log  cot  a  =  -f  0.143  16;  18.  log  cos  a  =  +  9.218  67; 

8.  log  cot  a  —  —  0.143  16;  19.  log  cos  a  =  —  9.218  67; 

9.  log  tan  a  =  -  9.024  81;  20.  log  tan  a  =  -  9.136  90; 

10.  log  tan  a  =  -  0.975 19;  21.  log  tan  a  =  -f-  9.136  90; 

11.  log  tan  a  =  +  0.975  19;  22.  log  cot  a  =  +  9.136  90. 


18.  Cases  when  the  Function  is  very  Small  or  Great. 

When  the  angle  of  which  we  are  to  find  the  functions  approaches 
to  zero,  the  logarithms  of  the  sine,  tangent,  and  cotangent  vary  so 
.rapidly  that  their  values  to  five  figures  cannot  be  readily  interpolated. 
The  same  remark  applies  to  the  cosine,  cotangent,  and  tangent  of 
angles  near  90°  or  270°.  The  mode  of  proceeding  in  these  cases  will 
depend  upon  circumstances. 

In  the  use  of  five-place  logarithms,  there  is  little  advantage  in 
carrying  the  computations  beyond  tenths  of  minutes,  though  the 
hundredths  may  be  found  when  the  tangent  or  cotangent  is  given. 
Where  greater  accuracy  than  this  is  required,  six-  or  seven-place 
tables  must  be  used. 

If  the  angles  are  only  carried  to  tenths  of  minutes,  there  is  no 
necessity  for  taking  out  the  sine,  tangent,  or  cotangent  to  more  than 
four  decimals  when  the  angle  is  less  than  3°,  and  three  decimal? 
suffice  for  angles  less  than  30'.  The  reason  is  that  this  number  of 
decimals  then  suffice  to  distinguish  each  tenth  of  minute. 

When  the  decimals  are  thus  curtailed,  an  expert  computer  will  be 
able  to  perform  the  multiplication  and  division  for  the  tenths  o? 
minutes  mentally.  If,  however,  this  is  inconvenient,  the  following 
rule  may  be  applied. 

To  find  the  log  sine  or  log  tangent  of  an  angle  less  than  2°  to 
four  places  of  decimals: 

EULE.    Enter  the  table  of  logarithms  of  numbers  with  the  valu* 


WHEN  THE  FUNCTION  IS  VERY  SMALL  OR  GREAT.        39 

of  the  angle  expressed  in  minutes  and  tenths,  and  take  out  the  loga- 
rithm. 

To  this  logarithm  add  the  quantity  6.4637. 
The  sum  will  be  the  log  sine,  and  the  log  tangent  may  be  assumed 
to  have  the  same  value. 

Example  1.     To  find  log  sin  1°  2$'.  6. 
1°  22'.  6  =  82'.6 

log  82'.6  =  1.9170 
constant,      6.4637 

log  sin  1°  22'. 6,      8.3807 

This  rule  is  founded  on  the  theorem  that  the  sines  and  tangents 
of  very  small  arcs  may  be  regarded  as  equal  to  the  arcs  themselves. 
Since,  in  using  the  trigonometric  functions,  the  radius  of  the  circle 
is  taken  as  unity,  an  arc  must  be  expressed  in  terms  of  the  unit 
radius  when  it  is  to  be  used  in  place  of  its  sine  or  tangent.  Now,  it 
is  shown  in  trigonometry  that  the  unit  radius  is  equal  to  57°. 2958  or 
3437'.  747  or  206  264".  8.  Hence  we  must  divide  the  number  of 
angular  units  in  the  angle  by  the  corresponding  one  of  these  coef- 
ficients to  obtain  the  length  of  the  corresponding  arcs  in  unit 
radius.  Now, 

log  3437. 747  =  3.5363 

co-log 6.4637 

which  may  be  added  instead  of  subtracting  the  logarithm. 

To  find  the  cosine  of  an  angle  very  near  90°,  we  find  the  sine  of 
its  complement,  which  will  then  be  a  very  small  angle,  positive  or 
negative. 

EXERCISES. 

Find  to  four  places  of  decimals: 

1.  log  sin         22'.  73; 

2.  log  sin    1°    1M2; 

3.  log  cos  90°    0'.78; 

4.  log  tan  88°  59'. 35; 

5.  log  cot  90°  28'.  76; 

6.  log  cos  89°  22'.23; 

7.  log  sin    0°    0'.25. 

If  an  angle  corresponding  to  a  given  sine  or  tangent  is  required, 
the  rule  is: 

From  the  given  log  sine  or  tangent  subtract  6.4637  or  add  3.5363. 
The  result  is  the  logarithm  of  the  number  of  minutes. 

Of  course  this  rule  applies  only  to  angles  less  than  2°,  in  the 
value  of  which  only  tenths  of  minutes  are  required. 


40  LOGARITHMIC  TABLES. 

EXERCISES. 
Find  a  from: 

1.  log  sin  a  =  7.2243;  3.  log  tan  a  =  -  3.8816; 

2.  log  cot  a  =  2.8816;  4.  log  cos  a  =        6.9218. 
When  the  small  angle  is  given  in  seconds.     Although  the  com, 

puter  may  take  out  his  angles  to  tenths  of  minutes,  cases  often  arise 
in  which  a  small  angle  is  given  in  seconds,  or  degrees,  minutes,  and 
.seconds,  and  in  which  the  trigonometric  function  is  required  to  five 
decimals.  In  this  case  the  preceding  method  may  not  always  give 
= accurate  results,  because  the  arc  and  its  sine  or  tangent  may  differ  by 
.a  greater  amount  than  the  error  we  can  admit  in  the  computation. 

Table  III.  is  framed  to  meet  this  case.  The  following  are  the 
•quantities  given: 

In  the  second  column :  The  argument,  in  degrees  and  minutes,  as 
•already  explained  for  Table  IV. 

In  the  first  column :  This  argument  reduced  to  seconds.  From 
this  column  the  number  of  seconds  in  an  arc  of  less  than  2°,  given  in 
degrees,  minutes,  and  seconds,  may  be  found  at  sight. 

Example.  How  many  seconds  in  1°  28'  39' ?  In  the  table,  before 
1°  28',  we  find  5280*,  which  being  increased  by  39"  gives  5319",  the 
number  required. 

Col.  3.  The  logarithm  of  the  sine  of  the  angle.  This  is  the  same 
•as  in  Table  IV. 

Col.  4.  The  value  of  log  sine  minus  log  arc;  that  is,  the  difference 
between  the  logarithm  of  the  sine  and  the  logarithm  of  the  number 
of  seconds  in  the  angle. 

Col.  5.  The  same  quantity  for  the  tangent. 

Cols.  6  and  7.  The  complements  of  the  preceding  logarithms,  dis- 
tinguished by  accents. 

The  use  of  the  tables  is  as  follows. 

To  find  the  sine  or  tangent  of  an  angle  less  than  2°: 

Express  the  angle  in  seconds  ~by  the  first  two  columns  of  the  table. 

Write  down  the  logarithm  in  column  8  or  column  T,  according  as 
the  sine  or  a  tangent  is  required. 

Find  from  Table  I.  the  logarithm  of  the  number  of  seconds. 

Adding  this  logarithm  to  S  or  T,  the  sum  will  be  the  log  sine  or 
log  tangent. 

Example.    Find  log  sin  1°  2'  47'.9. 

8,  4.68555 
1°  2'  47*. 9  =  3767*.9;  log,  3.576  10 

log  sin  1°  2'  47*.9,  8.261  65 


WHEN  THE  FUNCTION  IS  VERY  SMALL  OR  GREAT.   4J 

To  find  the  arc  corresponding  to  a  given  sine  or  tangent: 

Find  in  the  column  L.  sin.  the  quantity  next  greater  or  next 
smaller  than  the  given  logarithm. 

Take  the  corresponding  value  of  S'  or  T'  according  as  the  given 
function  is  a  sine  or  tangent,  and  add  it  to  the  given  function. 

The  sum  is  the  logarithm  of  the  number  of  seconds  in  the  required 
angle. 

Example.     Given  log  tan  x  =  8.401  25,  to  find  x. 
log  tan  x,  8.401  25 
T',  5.31433 


logz,  3.71558 
x  =  5194'.  9  =  1°  26'  34'.  9,  from  col.  2. 

EXERCISES. 

Find:  1.  log  sin  0°  20'  20'.25; 

2.  log  tan  0°    0'    1'.2273; 

3.  log  sin  1°  59' 22'. 7; 

4.  log  tan  1°    0'59'.7. 
Find  x  from: 

1.  log  tans  =  8.42796; 

2.  log  tan  x  =  7.42796; 

3.  log  tan  x  =  6.427  96; 

4.  log  sin  x  =  5.35435; 

5.  log  sin  x  =  4.226  19; 

6.  log  sin  x  =  8.540  78. 

When  the  cosine  or  cotangent  of  an  angle  near  90°  or  270°  is  re- 
quired, we  take  its  difference  from  90°  or  270°,  and  find  the  comple- 
mentary function  by  the  above  rules. 

Remark.  The  use  of  the  logarithms  of  the  trigonometric  func- 
tions is  so  much  more  extensive  than  that  of  the  functions  themselves 
that  the  prefix  "log"  is  generally  omitted  before  the  designation  of 
the  logarithmic  function,  where  no  ambiguity  will  result  from  the 
omission. 


TABLE  V. 
NATURAL  SINES  AND  COSINES. 


19.  This  table  gives  the  actual  numerical  values  of  the  sine  and 
cosine  for  each  minute  of  the  quadrant. 

To  find  the  sine  or  cosine  corresponding  to  a  given  angle  less  than 
45°,  we  find  the  degrees  at  the  top  of  a  pair  of  columns  and  the 
minutes  on  the  left. 

In  the  two  columns  under  the  degrees  and  in  the  line  of  minutes 
we  find  first  the  sine  and  then  the  cosine,  as  shown  at  the  head  of 
the  column. 

A  decimal  point  precedes  the  first  printed  figure  in  all  cases,  ex- 
cept where  the  printed  value  of  the  function  is  unity. 

If  the  given  angle  is  between  45°  and  90°,  find  the  degrees  at  the 
bottom  and  the  minutes  at  the  right. 

Of  the  two  numbers  above  the  degrees,  the  right-hand  one  is  the 
sine  and  the  left-hand  one  the  cosine. 

For  angles  greater  than  90°  the  functions  are  to  be  found  ac- 
cording to  the  precepts  given  in  the  case  of  the  logarithms  of  the 
sines  and  tangents. 


TABLE  VL 
ADDITION  AND  SUBTRACTION  LOGARITHMS. 


20.  Addition  and  subtraction  logarithms  are  used  to  solve  the 
problem:  Having  given  the  logarithms  of  two  numbers,  to  find  the 
logarithm  of  the  sum  or  difference  of  the  numbers. 

The  problem  can  of  course  be  solved  by  finding  the  numbers 
corresponding  to  the  logarithms,  adding  or  subtracting  them,  and 
taking  out  the  logarithm  of  their  sum  or  difference.  The  table 
under  consideration  enables  the  result  to  be  obtained  by  an  abbrevi- 
ated process. 

I.  Use  in  addition.  The  principle  on  which  the  table  is  con- 
structed may  be  seen  by  the  following  reasonings.  Let  us  put 

8=*a  +  b, 

a  and  b  being  two  numbers  of  which  the  logarithms  are  given.     "We 
shall  have 


putting,  for  clearness,  x  =  -. 

We  then  have 

log  S  =  log  a  +  log  (1  +  x). 

Since  log  a  and  log  b  are  both  given,  we  can  find  log  x  from  the 
equation 

log  x  =  log  b  —  log  ay 
which  is  therefore  a  known  quantity. 

Now,  for  every  value  of  log  x  there  will  be  one  definite  value  of 
each  of  the  quantities  x,  1  +  #>  and  log  (I-{-  x).  Therefore  a  table 
may  be  constructed  showing,  for  every  value  of  log  x,  the  correspond- 
ing value  of  log  (1  -f-  x). 

Such  a  table  is  Table  VI. 

The  argument,  in  column  A,  being  log  x,  the  quantity  B  in  the 
table  is  log  (1  -j-  x). 

Example,    log  0.25  =  9.  397  94. 

Entering  the  table  with  A  =  9.397  94,  we  find 

J3  =  0.096  91, 
which  is  the  logarithm  of  1.25. 


44  LOGARITHMIC  TABLES. 

Therefore,  entering  the  table  with  log  x  as  the  argument,  we 
take  out  log  (1  -}-  x),  which  added  to  log  a  will  give  log  S. 

We  have  therefore  the  following  precept  for  using  the  table  in 
addition: 

Take  the  difference  of  the  two  given  logarithms. 

Enter  the  table  with  this  difference  as  the  argument  A,  and  take 
out  the  quantity  B. 

Adding  B  to  the  subtracted  logarithm,  the  sum  will  be  the  required 
logarithm  of  the  sum. 

It  is  indifferent  which  logarithm  is  subtracted,  but  convenience 
in  interpolating  will  be  gained  by  subtracting  the  greater  logarithm 
from  the  lesser  increased  by  10.  The  number  B  will  then  be  added 
to  the  greater  logarithm. 

Example.  Given  log  m  =  1.62974,  log  n  =  2.203  86  ;  find, 
log  (m  +  n). 

The  required  logarithm  is  found  in  either  of  the  following  two 

ways: 

Jog  m,  1.629  74  (1)  log  £,  0.676  76  (4) 

log  'n,  2.203  86  (2)  log  m,  1.629  74  (1) 

B,  0.102  64  (4)  log  »,  2.203  86  (2) 

A  =  log  w  -r-  n,  9.425  88     (3)  log  n  -=-  ra,  0.574  12     (3) 

log  (m  +  n),  2.306  50     (5)          log  (m  +  »),  2.306  50     (5) 

The  figures  in  parentheses  show  the  order  in  which  the  numbers 
are  written. 

EXERCISES. 
Log  a  and  log  b  having  the  following  values,  find  log  (a  -f-  b). 

1.  log  a  =  1.700  37;   log  b  =  0.921  69. 

2.  log  a  =  0.624  60;   log  b  =  9.881  26. 

3.  log  a  =  9.791  86;   log  b  =  9.322  09. 

4.  log  a  =  1.601  62;   log  I  =  1.306  06. 

5.  log  a  =  0.792  90;   log  b  =  9.221  27. 

6.  log  a  =  0.601  32;   log  b  =  9.001  68. 

7.  log  a  =  4.796  43;   log  b  =  3.981  86. 

II.  Use  in  subtraction.  The  problem  is,  having  given  log  a  and 
log  I,  to  find  the  logarithm  of 

D  -  a  -  b. 
Wehave 


ADDITION  AND  SUBTRACTION  LOGARITHMS.  45 


Since  log  -r  is  found  by  subtracting  log  b  from  log  a,  if  we  can 

find  log  IT-  —  l)  from  log  j-,  the  problem  will  be  solved. 

From  the  construction  of  the  table  already  explained,  if  we  have 

we  must  have 


We  now  have  the  following  precept  for  subtraction: 
Subtract  the  lesser  of  the  given  logarithms  from  the  greater. 
Enter  the  table  so  as  to  find  the  difference  of  the  logarithms  in  the 
numbers  B  of  the  table. 

Add  the  corresponding  value  of  A  to  the  lesser  of  the  given  loga- 
rithms.    The  sum  will  be  the  logarithm  of  the  difference. 
.     Example.     Find  log  (n  —  m)  in  the  example  of  the  preceding 
section. 

log  n,  2.203  86     (1) 

log  m,  1.62974    (2) 

^,0.43945     (4) 

log—  =  B,  0.57412     (3) 
m 

log  (n  -  m),  2.069  19     (5) 

EXERCISES. 

Find  the  logarithms  of  the  differences  of  the  quantities  a  and  b 
in  the  preceding  section. 

Remark.  In  the  use  of  addition  and  subtraction  logarithms, 
the  precepts  apply  to  numerical  sums  and  differences,  without 
respect  to  the  algebraic  signs  of  the  quantities.  For  example,  the 
algebraic  difference  between  -f-  1473  and  —  29  462  is  to  be  found  by 
addition,  and  the  algebraic  sum  of  a  positive  and  negative  quantity 
by  subtraction. 

Case  where  the  quotient  is  large.  Near  the  end  of  the  table,  A 
and  B  become  nearly  equal;  the  structure  of  the  table  is  therefore 
changed  so  as  to  simplify  its  use.  It  is  evident  that  if  b  is  very 
small  compared  with  a,  the  logarithms  of  a  -f-  b  and  a  —  b  will 
not  differ  much  from  the  logarithm  of  a  itself.  Hence,  in  this  case, 
we  shall  have  smaller  numbers  to  use  if  we  can  find  the  quantity 
which  must  be  added  to  log  a  to  give  log  (a  -f-  b),  or  subtracted  from 


46  LOGARITHMIC  TABLES. 

log  a  to  give  log  (a  —  5).     Now,  the  equations  already  written  give, 
when  a  >  b,  log  a  =  log  ft  -{-  A, 

log  (a  +  1)  =  log  b  +  £; 
whence,  by  subtraction, 

log  (a  +  b)  —  log  a  =  B  —  A, 

or  log  (a  +  b)  =  log  a  +  £  -  A.     (with  Arg.  ^) 

We  find  in  the  same  way, 

log  (a  -  b)  =  log  a  -  (B  -  A),  (with  Arg.  B) 
Now,  whenever  log  a  —  log  ~b  is  greater  than  1.65,  we  shall  find 
it  more  convenient  to  take  out  B  —  A  from  the  table  than  either 
A  or  B.  We  notice  that  the  last  two  figures  of  B  in  this  part  of 
the  table  vary  slowly,  and  we  need  only  attend  to  them  in  interpolate 
ing.  For  instance,  in  the  horizontal  line  corresponding  to  A  =  1.66 
we  find: 

for  A  =  1.660  00;     B  =  1.669  40;    B  -  A  =  .009  40; 
.66100;  .67038;  .00938; 

.66200;  .67136;  .00936; 

.66300;  .67233;  ,00933; 

.66400;  .67331;  .00931; 

.66500;  .67429;  .00929; 

etc.  etc.  etc. 

The  interpolation  of  B  —  A  is  now  very  easy  whether  the  quan- 
tity given  is  A  or  B.  We  note  that  B  —  A  has  but  three  significant 
figures,  of  which  the  first  is  found  in  column  zero,  and  the  other  two 
are  the  last  two  figures  of  B  as  printed. 

As  an  example,  let  us  find  log  (a  +  b)  from 
logfl  =  2.79163 
log£  =  1.12819 

A  =  1.66344 

Entering  the  table  with  this  value  of  A,  we  find  by  column  0 
that  B  —  A  falls  between  .009  40  and  .009  19.  Following  the  hori- 
zontal line  A  =  1.66  to  column  3  and  interpolating  the  last  two 
figures  between  33  and  31  for  .44,  with  the  difference  —  2,  we  find 

B  -  A  =    .00932 
Then  log  a  =  2. 791  63 

log(0  +  b)  =  2.80095 

Next,  if  log  (a  —  b)  is  required,  we  have  to  find  the  difference 
1.663  44  in  the  part  B  of  the  table.     We  find  in  the  table: 
for  B  =  1.662  55;    B  -  A  =  .009  55; 
for  B  =  1.663  53;    B  -  A  =  .009  53. 


ADDITION  AND  SUBTRACTION  LOGARITHMS.  47 

Therefore 

for  B  =  1.663  44;    B  -  A  =  .009  53. 

Subtracting  this  from  log  a,  we  have 

log  (a  -  b)  =  2.78210. 

EXEECISES. 

^'nd  log  (a  +  £)  and  log  (a  —  £)  from: 

8.  log  a  =  0.367  02;   log  b  =  8.462  83. 

9.  log  a  =  0.001  26;   log  b  =  8.329  07. 

10.  log  a  =  2.069  23;   log  b  =  0.110  85. 

11.  log  a  =  5.807  35;   log  b  =  3.83809. 

For  values  of  A  and  B  greater  than  2.00,  the  table  is  so  arranged 
that  no  interpolation  at  all  is  necessary.  The  computer  has  only  to 
find  what  value  of  A  or  B  given  in  the  table  comes  nearest  his  value 
of  log  a  —  log  b  and  take  the  corresponding  value  of  B  —  A.  He 
must  remember  that  column  A  is  to  be  entered  for  addition,  and  B 
for  subtraction. 

In  this  part  of  the  table  A  and  B  are  given  to  fewer  than  five 
decimals;  because  five  decimals  are  not  necessary  to  give  B  —  A  with 
accuracy.  The  nearer  the  end  of  the  table  is  approached,  the  fewer 
the  decimals  necessary  in  taking  the  difference. 

Example.    Find  log  (a  -f  b)  and  log  (a  —  b)  from 
log  a  =  1.265  32 
log  b  =  9.22230 

log  a  —  log£,  2.04302 

Entering  column  A  with  this  difference,  we  find  the  nearest  tabu- 
lar value  of  A  to  be  2.0425,  to  which  corresponds  B  —  A  =  .003  92. 
Hence 

log  (a  +  b)  =  1.265  32  -f  .003  92  =  1.269  24. 
Entering  column  B  with  the  same  difference,  we  find  B  — •  A  SB  ' 
.00395;  whence 

log  (a  -  b)  =  1.265  32  -  .003  95  =  1.261 37. 

EXERCISES. 
Find  log  (a  -f-  b)  and  log  (a  —  b)  from: 

1.  log  a  =  4.069  05;  log  b  =  2.001  32. 

2.  log  a  =  3.926  93;  log  b  =  1.201  59. 

3.  log  a  =  3.061  64;  log  b  =  0.126  15. 

4.  log  a  =  1.22C  68;  log  b  =  7.321  56. 

5.  log  a  =  0.693  17;  log  5  =  6.010  23. 

6.  log  a  =  2.306  20;  log  b  =  7.023  01. 


48  LOGARITHMIC  TABLES. 

Case  of  nearly  equal  numbers.  Near  the  beginning  of  the  table 
the  reverse  is  true:  it  is  not  possible  to  find  A  with  accuracy  to  five 
places  of  decimals.  But  here  the  value  of  A  taken  from  the  tables, 
though  it  be  found  to  only  two,  three,  or  four  places  of  decimals, 
will  give  as  accurate  a  result  as  the  computation  of  a  and  b  to  five 
places  will  admit  of.  Let  us  suppose,  for  example,  that  we  have  to 
find  log  (a  —  b)  from 

log  a  =  9.883  15 

log  b  =  9.88296 

B  =  0.00019 

We  find  ^4  =  6.64-10; 

whence  log  (a  —  b)  =  6.52  —  10. 

We  note  that  the  value  of  A  may  be  6.63  or  6.65  as  well  as  6.64, 
so  that  the  result  cannot  be  carried  beyond  two  decimals.  To  show 
that  these  two  are  as  accurate  as  the  work  admits  of*  we  find  the 
natural  numbers  a  and  b  from  Table  I. 

a=  0.76410 
b  =  0.76377 

a-b  =  0.00033 

Since  a  —  b  has  but  two  significant  figures,  and  the  first  of  these 
is  less  than  5,  two  figures  in  the  logarithm  are  all  that  can  be 
accurate. 


TABLE  YIL 
SQUARES  OF  NUMBERS. 


21.  By  means  of  this  table  the  square  of  any  number  less  than 
1000  may  be  found  at  sight,  and  that  of  any  number  less  than  10  000 
by  a  simple  and  easy  interpolation. 

The  first  page  gives  the  squares  of  the  first  100  numbers,  which 
it  is  often  convenient  to  have  by  themselves. 

On  the  second  and  third  pages  (98  and  99)  the  hundreds  of  the 
number  to  be  squared  are  found  at  the  tops  of  the  several  columns, 
and  the  tens  and  units  in  the  left-hand  column.  The  first  three  or 
four  figures  of  the  square  are  in  the  column  under  the  hundreds, 
and  opposite  the  tens  and  units,  and  the  last  two  figures  on  the  right 
of  the  page  after  the  column  9  +  + 

Examples.  The  square  of  634  is  401  956; 
"  "  329  "  108241; 
"  "  265  "  70225; 
"  "  153  "  23409; 
«  «  999  "  998001. 

The  same  table  may  be  used  for  any  number  of  three  significant 
figures  by  attention  to  the  position  of  the  decimal-point.     Thus: 
511009  =  2611210000; 

511s         =      261121; 
51.1'      =          2611.21; 
5.11'    =  26.1121; 

0.5119  =  0.261121. 

When  there  are  four  significant  figures,  an  interpolation  may  be 
executed  in  several  ways.  If  n  be  the  nearest  number  the  square  of 
^hich  is  found  in  the  table,  and  h  the  excess  of  the  given  number 
over  this,  so  that  n  -{•  his  the  number  whose  square  is  required,  we 
shall  have 


where  N  =  n  +  h9  the  given  number. 


50  LOGARITHMIC  TABLES. 

We  may  therefore  find  the  square  of  257.4  in  the  following  way: 

257s  =  66  049 
514.4  X  .4  =       205.76 

(257. 4)2  =  66254.76 

To  find  the  square  of  9037  we  proceed  thus: 
9037 
9030a          =  81  540  900 

18067  X  7  =       126469 
9037a         =  81667369 

In  many  cases  only  one  more  figure  will  be  required  in  the  square 
than  in  the  given  number.  The  square  can  then  be  interpolated  with 
all  required  accuracy  by  the  differences,  the  last  two  figures  of  which 
are  found  in  the  last  column  of  the  table,  while  the  remaining  figures 
are  found  by  taking  the  difference  between  two  consecutive  numbers 
in  the  principal  column. 

To  return  to  the  last  example,  we  find  the  difference  between 
257a  and  2583  to  be  515,  the  first  figure  being  the  difference  between 
660  and  665,  and  the  last  two,  15,  in  the  last  column.  Then 

2573  =  66  049 
515  X  0.4  =       206 

(257. 4)a  =  66255 
—which  is  correct  to  the  nearest  unit. 

It  will  be  remarked  that  the  two  methods  are  substantially  the 
same  when  only  five  figures  are  sought  in  the  result.  The  substantial 
identity  rests  upon  the  general  theorem  that 

The  difference  of  the  squares  of  two  consecutive  numbers  is  equal 
to  the  sum  of  the  numbers. 

We  prove  this  theorem  thus: 

(n  +  l)a  -  n*  =  2n  +  1  =  n  +  (n  +  1). 

When  the  tabular  difference  is  taken  in  the  way  already  described, 
it  will  often  happen  that  the  difference  between  the  numbers  in  the 
columns  of  hundreds  is  to  be  diminished  by  unity.  Thus,  although 
4173  —  4160  =  13,  the  difference  between  645s  and  6462  is  not  1391, 
but  1291.  These  cases  are  noted  by  the  asterisk  after  the  number  in 
the  last  column. 

The  squares  of  numbers  of  more  than  four  figures  may  be  found 
in  the  same  way,  but  in  such  cases  it  will  generally  be  easier  to  use 
logarithms  than  the  table  of  squares. 


TABLE  VIII. 

TO   CONVERT  HOURS,   MINUTES,  AND  SECONDS 
INTO  DECIMALS  OF  A  DAY,  AND  VICE  VEKSA. 


.  The  familiar  method  of  solving  this  problem  is  to  convert 
the  seconds  into  decimals  of  a  minute,  and  the  minutes  into  decimals 
of  an  hour,  by  dividing  by  60,  and  then  the  hours  into  decimals  of  a 
day  by  dividing  by  24.  The  reverse  problem  is  solved  by  multiply- 
ing by  24,  60,-  and  60. 

Table  VIII.  enables  us  to  perform  these  operations  without  divi- 
sion. Column  D  gives  each  hundredth  of  a  day,  but  its  numbers  may 
also  be  regarded  as  ten  thousandths  or  millionths  of  a  day,  according 
to  which  of  the  following  three  columns  is  used.  In  column  H.M.S. 
are  found  the  hours,  minutes,  and  seconds  corresponding  to  these 
hundredths.  In  the  next  column  is  one  hundredth  of  column  H.  M.  S.9 
or  the  minutes  and  seconds  in  the  number  of  ten  thousandths  of  a 

day  in  column  D.     Finally,  column     '     '  —  shows  the  number  of 

iuu 

seconds  in  the  number  of  millionths  of  a  day  found  in  column  D. 
Example.  To  convert  Od.532  946  into  hours,  minutes,  and  seconds. 
Od.53  =  12h  43m  12s 

.002  9      =          4m  108.56 
.000046=  38.97 


Od.532  946  =  12h  47m  268.53 

It  will  be  seen  that  we  divide  the  figures  of  the  given  decimal  of 
a  day  into  pairs,  and  enter  the  three  columns  of  time  with  these 
three  pairs  in  succession. 

If  seven  decimals  are  given,  we  may  interpolate  the  last  number, 
as  in  taking  out  a  logarithm. 

Example.     Convert  Od.050  762  7. 

Od.05  =  lh  12m  O8 

.000  7  =        lm  0s. 48 

.000062  =  5s.  36 

.0000007  =  .7X.08=  0-.06 


lh  13m  58.90 


52  LOGARITHMIC  TABLED 

In  practice  the  computer  will  perform  the  interpolation  mentally, 
adding  .7  X  .08  =  .06  to  the  number  5.36  of  the  table  in  his  head, 
and  writing  down  5s. 42  as  the  last  quantity  to  be  added. 

EXERCISES. 

Convert  into  hours,  minutes,  and  seconds: 

1.  Od.2030792; 

2.  Od.  783  605  8; 

3.  Od.0102034; 

4.  Od.  990  990  9. 

To  use  the  table  for  the  reverse  operation,  we  proceed  as  in  the 
following  example: 

It  is  required  to  convert  17h  29m  30s.  93  into  decimals  of  a  day. 
Looking  in  the  table,  we  find  that  the  required  decimal  is  between 
0.72  and  0.73.  Hence  the  first  two  figures  are  0.72,  the  equivalent 
of  17h  16m  48s.  Subtracting  the  lat-  1711  ggm  30*.  93 

ter   from    the    given    number,    we      0.72  =  17h  16m  48s 

have  a  remainder  12m  428.93,  to  be  12m  42s.  93 

,,.     .        .          H.M.S.      _.          -0088        =        12m  40s. 32 
sought  for  in  column  —m~.    This        >000  030  2  =  "  ~2^61 

gives  88  as  the  next  two  figures.     Subtracting  the  equivalent  of 
.0088  or  12m  408.32,  we  have  left  2s.  61,  which  we  are  to  seek  in 

TT   -»r   nr 

column  — '     '   '.  We  find  the  corresponding  number  of  column  D  to 

100 

be  302.     Hence 

17h  29m  308.93  =  Od.  728  830  2. 

In  solving  this  problem  the  computer  should  be  able,  after  a  little 
practice,  to  perform  the  subtractions  and  carry  the  remainders  men- 
tally, thus  saving  himself  the  trouble  of  writing  down  the  numbers. 

EXERCISES. 

Take  the  answers  obtained  from  the  four  preceding  exercises, 
subtract  each  result  from  24h  Om  0B,  change  the  remainder  to  deci- 
mals of  a  day,  and  see  if  when  added  to  the  decimals  of  the  preceding 
exercises  the  sum  is  ld.  000  000  0,  as  it  should  be. 


TABLE  IX. 
TO  CONVERT  TIME  INTO  ARC,  AND   VICE  VERSA. 


23.  In  astronomy  the  right  ascensions  of  the  heavenly  bodies 
are  commonly  given  in  hours,  minutes,  and  seconds,  the  circumfer- 
ence being  divided  into  24  hours,  each  hour  into  60  minutes,  and 
each  minute  into  60  seconds. 

Since  360r  =  one  circumference, 

ve  have  lh  =  15°; 

lm  =  15'; 

1s   =  15'; 

the  signs  h,  m,  and  8  indicating  hours,  minutes,  and  seconds  of  time. 

Hence  we  may  change  time  into  arc  by  multiplying  by  15,  and 
arc  into  time  by  dividing  by  15,  the  denominations  being  changed  in 
each  case.  Table  IX.  enables  us  to  do  this  by  simple  addition  and 
subtraction  by  a  process  similar  to  that  employed  in  changing  hours, 
minutes,  and  seconds  into  decimals  of  a  day. 

To  turn  time  into  arc,  we  find  in  the  table  the  whole  number  of 
degrees  contained  in  the  time  denomination  next  smaller  than  the 
given  one,  and  subtract  the  former  time  denomination  from  the 
latter. 

Next  we  find  the  minutes  of  arc  corresponding  to  the  given  time 
next  smaller  than  the  remainder,  and  again  subtract. 

Lastly  we  interpolate  the  seconds  corresponding  to  the  second 
remainder. 

Example.     Change  15h  29m  468.24  to  arc. 

Given  time,  15h  29m  468.24 

The  table  gives  232°  =  15h  28m 

Remainder,  lm  468.24 

The  table  gives      26'  =  lm  448 


Remainder,  2s.  24  =  33'.  6 

Hence 

15h  29m  468.24  =  232°  26'  33'.  6. 


54  LOGARITHMIC  TABLES. 

The  computer  should  be  able  to  go  through  this  operation  with- 
out writing  down  anything  but  the  result. 

The  operation  of  changing  arc  into  time  is  too  simple  to  require 
description,  but  it  is  more  necessary  to  write  down  the  work. 

EXERCISES. 

Change  the  following  times  to  arc,  and  then  check  the  results  by 
changing  the  arcs  into  time  and  seeing  whether  the  original  times 
are  reproduced: 

1.  7h  29m  178.86; 

2.  Oh     4m     08.25; 

3.  12h    4m    0s.  25; 

4.  13h  48m  169.40; 

5.  19h    7m  598.92. 


TABLE  X. 

TO  CONVERT  MEAN  TIME  INTO  SIDEREAL  TIME, 
AND  SIDEREAL  INTO  MEAN  TIME. 


24.  Since  365£  solar  days  =  366^-  sidereal  days  (very  nearly),, 
any  period  expressed  in  mean  time  may  be  changed  to  sidereal  time- 

by  increasing  it  by  its  -          part,  and  an  interval  of  sidereal  time- 

uDO./oO 

may  be  changed  to  mean  time  by  diminishing  it  by  its  -  part.. 

ODD.  -CO 

The  first  part  of  the  table  gives,  for  each  10  minutes  of  the  argu- 
ment, its  Q          part,  by  which  it  is  to  be  increased.     The  second: 


part  of  the  table  gives  the  O^FITH  Par^  °f  the  argument. 


The  small  table  in  the  margin  shows  the  change  for  periods  of 
less  than  10  minutes. 

Example  1.     To  change  17h  48m  36s.  7  of  mean  time  to  sidereai 

time. 

Given  mean  time,        17h  48m  36s.  70 

Corr.  for  17h  40m,  2m  54M3 

Corr.  for          8m  37',  18.41 

Sidereal  time,  17h  51m  32s.  24 

Ex.  2.     To  change  this  interval  of  sidereal  time  back  to  mean 
time. 

Corr.  for  17h  50m,  -  2m  55".  29 

Corr.  for  lm  32%     -          08.25 

—  2m  558.54 
Sidereal  time,  17h  51m  328.24 

Mean  time,  17h  48m  368.70 

EXERCISES. 
Change  to  sidereal  time: 

1.  3h  42m  36".  5  m.  t.;        3.     22h     3m    5§.61  m.  t* 

2.  18h  46m  298.82     "  4.       Oh     lm  12B.55      " 
Change  to  mean  time: 

5.  Oh     7m  168.3  sidereal  time; 

6.  22h  17m  298.65  " 


56  OF  INTERPOLATION. 


OF  DIFFERENCES  AND  INTERPOLATION.* 


25.  General  Principles. 

"We  call  to  mind  that  the  object  of  a  mathematical  table  is  to 
enable  one  to  find  the  value  of  a  function  corresponding  to  any  value 
whatever  of  the  variable  argument.  Since  it  is  impossible  to  tabulate 
the  function  for  all  values  of  the  argument,  we  have  to  construct  the 
table  for  certain  special  values  only,  which  values  are  generally  equi- 
distant. For  example,  in  the  tables  of  sines  and  cosines  in  the 
present  work  the  values  of  the  functions  are  given  for  values  of  the 
argument  differing  from  each  other  by  one  minute. 

The  process  of  finding  the  values  of  functions  corresponding  to 
values  of  the  argument  intermediate  between  those  given  is  called 
interpolation. 

We  have  already  had  numerous  examples  of  interpolation  in  its' 
most  simple  form;  we  have  now  to  consider  the  subject  in  a  more 
general  and  extended  way. 

In  the  first  place,  we  remark  that,  in  strictness,  no  process  of 
interpolation  can  be  applicable  to  all  cases  whatever.  From  the 
mere  facts  that 

To  the  number  2  corresponds  the  logarithm  0.301  03, 
"    "        "        3         "  "         "         0.477  12, 

we  are  not  justified  in  drawing  any  conclusion  whatever  respecting 
"the  logarithms  of  numbers  between  2  and  3.  Hence  some  one  or 
more  hypotheses  are  always  necessary  as  the  base  of  any  system  of 
interpolation.  The  hypotheses  always  adopted  are  these  two: 

1.  That,  supposing  the  argument  to  vary  uniformly,  the  function 
varies  according  to  some  regular  law. 

2.  That  this  law  may  be  learned  from  the  values  of  the  function 
given  in  the  table. 

These  hypotheses  are  applied  in  the  process  of  differencing,  the 

*  The  study  of  this  subject  will  be  facilitated  by  first  mastering  so  much  of 
it  as  is  contained  in  the  author's  College  Algebra,  §§  299-302. 

It  is  also  recommended  to  the  beginner  in  the  subject  that,  before  going 
over  the  algebraic  developments,   he  practise  the  methods  of  computation 
according  to  the  rules  and  formulae,  so  as  to  have  a  clear  practical  understand 
ing  of  the  notation.     He  can  then  more  readily  work  out  the  developments. 


GENERAL  PRINCIPLES.  57 

nature  of  which  will  be  seen  by  the  following  example,  where  it  is 
applied  to  the  logarithms  of  the  numbers  from  30  to  37: 

Function.          A'        4"     A"<    A" 
log  30.     1.47712 

" 


31.  1.491  36         ~     •-  45    ,    „ 

"   32.  1.505  15  T  JoS  -  43  J  *  +  2 

"  33.  1.518  51  J  i9Q7  -  39  J  *  -  8 

«   34.  1.531  48  J  J*JJ  _  38  +  £  +  1 

«   35.  1.544  07  +  }f*  _  36  +  *  +  1 

"  36.  1.556  30  "tifS  -  33  +  ' 

"37.  1.568  20  ~* 

The  column  A'  gives  each  difference  between  two  consecutive 
values  of  the  function,  formed  by  subtracting  each  number  from  that 
next  following.  These  differences  are  called  first  differences. 

The  column  A"  gives  the  difference  between  each  two  consecu- 
tive first  differences.  These  are  called  second  differences. 

In  like  manner  the  numbers  in  the  succeeding  columns,  when 
written,  are  called  third  differences,  fourth  differences,  etc. 

Now  if,  in  continuing  the  successive  orders  of  differences,  we  find 
them  to  continually  become  smaller  and  smaller,  or  to  converge  to- 
ward zero,  this  fact  shows  that  the  values  of  the  functions  follow  a 
regular  law,  and  the  first  hypothesis  is  therefore  applicable. 

In  order  to  apply  interpolation  we  must  then  assume  that  the 
intermediate  values  of  the  function  follow  the  same  law.  The  truth 
of  this  assumption  must  be  established  in  some  way  before  we  can 
interpolate  with  mathematical  rigor,  but  in  practice  we  may  suppose 
it  true  in  the  absence  of  any  reason  to  the  contrary. 

26.  Effect  of  errors  in  the  values  of  the  functions.  In  the  pre- 
ceding example  it  will  be  noticed  that  if  we  continue  the  orders  of 
differences  beyond  the  fourth,  they  will  begin  to  increase  and  become 
irregular.  This  arises  from  the  imperfections  of  the  logarithms, 
owing  to  the  omission  of  decimals  beyond  the  fifth,  already  described 
in  §11. 

When  we  find  the  differences  to  become  thus  irregular,  we  must 
be  able  to  judge  whether  this  irregularity  arises  from  actual  errors  in 
the  original  numbers,  which  ought  to  be  corrected,  or  from  the  small 
errors  necessarily  arising  from  the  omission  of  decimals* 

The  great  advantage  of  differencing  is  that  any  error,  however 
small,  in  the  quantities  differenced,  unless  it  follows  a  regular  law, 
will  be  detected  by  the  differences.  To  show  the  reason  of  this,  we 
investigate  what  effect  errors  in  the  given  functions  will  have  upon 
the  successive  orders  of  differences. 


68  OF  INTERPOLATION. 

THEOREM.    The  differences  of  the  sum  of  two  quantities  are  equal 
to  the  sums  of  their  differences. 
General  proof.     Let 

/„  /„,  /„,  etc.,  be  one  set  of  functions; 
//,  /,',/,',  etc.,  another  set. 
fi  +//>  /a  +  //>  /s  +  /»'>  etc.,  wil1  then  be  their  sums. 

In  the  first  of  the  following  columns  we  place  the  first  differences 
of/,  in  the  second  those  of/',  and  in  the  third  those  of  /  +  /',  each 
formed  according  to  the  rule  : 


etc.  etc.  etc. 

It  will  be  seen  that  the  quantities  in  the  third  column  are  the 
sums  of  those  in  the  first  two. 

NUMERICAL  EXAMPLE. 
/       A  f      A'  f+f     A 

lt+n          l  +  i         ll  +  n 

*n  ~r  H  c  +  3  t-f.  +  14 

_5?-51  10  +  4  9~47 

We  see  that  the  third  set  of  values  of  Ar  follow  the  theorem. 
Because  the  second  differences  are  the  differences  of  the  first,  the 
third  the  differences  of  the  second,  etc.,  it  follows  that  the  theorem 
is  true  for  differences  of  any  order. 

Now  when  we  write  a  series  of  functions  in  which  the  decimals  ex- 
ceeding a  certain  order  are  omitted,  we  may  conceive  each  written  num- 
ber to  be  composed  of  the  algebraic  sum  of  two  quantities,  namely: 

1.  The  true  mathematical  value  of  the  function. 

2.  The  negative  of  the  omitted  decimals. 

Example.  In  the  preceding  collection  of  logarithms,  since  the 
true  value  of  log  30  is  1.477  121  3  .  .  .  ,  we  may  conceive  the  quantity 
written  to  be 

1.477  12  =  log  30  -  .000  001  3  ____ 

Hence  the  differences  actually  written  are  the  differences  of  the 
true  logarithms  minus  the  differences  of  the  errors.  Now  suppose 
the  errors  to  be  alternately  +  0.5  and  —  0.5  =  the  point  marking 
off  the  last  decimal.  Their  differences  will  then  be  as  follows: 

/'        J'       A"     4'" 

-  0.5    ,    x  +  2  __  4 
+  0.5         i-2 

-  0.5  ~  \  +  2  ' 
+  0.5  "     L  -  2  " 

etc.     etc.    etc.    etc. 


GENERAL  PRINCIPLES.  59 

It  is  evident  that  the  wth  order  of  differences  of  the  errors  are 
equal  to  ±  2"-1.  Hence,  in  this  case,  if  the  nth  order  of  differences 
of  the  true  values  of  the  function  were  zero,  still,  in  consequence  of 
the  omission  of  decimals,  the  actual  differences  of  the  nth  order  would 
be2»-1. 

This,  however,  is  a  very  extreme  case,  since  it  is  beyond  all  proba- 
bility that  the  errors  should  alternate  in  this  way.     A  more  probable 
average  example  will  be  obtained  by  supposing  a  single  number  to  have 
an  error  of  0.5,  while  the  others  are  correct.     We  shall  then  have: 
f       A1         4"         d'"         ^1T         ^T 
0  0          A      +  0-5       o 


__ 

0  o  '     +  0.5 

In  this  case  the  maximum  value  of  the  difference  of  the  nth  order 
is  1.5  in  the  differences  of  the  third  order,  3  in  those  of  the  fourth, 
5  in  those  of  the  fifth,  etc.  Its  general  expression  is 

1  n  (n  -  1)  (n  -  2)  ____  (n  -  s  -f  1) 

2  1.2.  3....  s 
where  n  is  the  order  of  differences,  and 


n 

n 

-1 

*  =  2 

or  — 

2 

according  as  n  is  even  or  odd.     Thus: 

A'    =1 

. 

2 

> 

'  -  *L 

2 
'  1 

= 

i; 

1 

3 

I/ 

^'"  =    g- 

"  1 

= 

.< 

&     -1 

4. 

3 

3. 

2 

*  1. 

2 

y 

1 

5. 

4 

5. 

2 

'  1. 

2  "~ 

, 

etc.  etc. 

This  being  about  the  average  case,  in  actual  practice  the  differ- 
ences may  be  two  or  three  times  as  great  without  necessarily  imply- 
ing an  error  greater  than  0.5  in  the  numbers  written. 

We  have  now  the  following  general  rule  for  judging  whether  a 
series  of  numbers  do  really  follow  a  uniform  law: 

Difference  the  series  until  we  reach  an  order  of  differences  in  which 
the  4*  and  —  signs  either  alternate  or  follow  each  other  irregularly. 


60  OF  INTERPOLATION. 

If  none  of  the  differences  of  this  order  expressed  in  units  of  the 
last  place  of  decimals  exceed  the  limit 

n  (n  —  1)  .  .  .  _.  (n  —  s  +  1) 
1. 2. 3  .... s 

— that  is,  the  value  of  the  largest  binomial  coefficient  of  the  nth  order — 
the  given  numbers  may  be  assumed  to  follow  a  regular  law,  and 
therefore  to  be  correct  to  a  unit  in  the  last  figure. 

If  some  differences  exceed  this  limit,  their  quotient  by  the  above 
binomial  coefficient  may  be  considered  to  show  the  maximum  error 
with  which  the  number  opposite  it  is  probably  affected. 

We  can  thus  detect  an  isolated  error  in  a  series  of  numbers  with 
great  certainty.  Suppose,  for  example,  an  error  of  2  in  some  number 
of  the  series.  Differencing  the  series  0,  0,  0,  2,  0,  0,  0,  we  shall 
find  the  four  largest  differences  of  the  fifth  order  to  be  —  10,  -j-  20, 
—  20,  -[-  10,  which  would  enable  us  to  hit  at  once  upon  the  erro- 
neous number  and  judge  of  the  magnitude  of  its  error. 

An  error  near  the  beginning  and  end  of  the  series  of  numbers  of 
which  the  differences  are  taken  cannot  be  detected  by  the  differences 
unless  it  is  considerable.  If,  for  instance,  the  first  or  last  number 
is  in  error  by  1,  the  error  of  each  order  of  differences  will  only  be  1, 
as  we  may  easily  see  by  the  following  example: 
/'  A'  A"  A'" 

°~  I  +  I  -  1  etc. 

It  is  only  in  those  differences  which  are  on  or  near  the  same  line 
as  the  numbers  which  are  magnified  in  the  way  we  have  shown.  But 
at  the  beginning  and  end  of  the  series  we  cannot  determine  theso 
differences. 

Examining  the  various  tables  of  differences,  we  see  that  n  numbers 
have  n  —  1  first  differences,  n  —  2  second  differences,  and  so  on,  the 
number  diminishing  by  1  with  each  succeeding  order.  Hence,  unless 
the  number  of  given  functions  exceeds  the  index  expressing  the  order 
of  differences  which  we  have  to  form,  no  certain  conclusion  can  be 
drawn. 

What  is  here  said  of  the  correctness  of  the  numbers  when  the 
differences  run  properly  must  be  understood  as  applicable  to  isolated 
errors  only.  If  all  the  numbers  were  subject  to  an  error  following  a 
regular  law,  this  error  would  not  be  detected  by  the  differences  be- 
cause, from  the  nature  of  the  case,  the  latter  only  show  deviations 
from  some  regular  law. 


FUNDAMENTAL  FORMULA.  61 


27.  Fundamental  Formulae  of  Interpolation. 

We  suppose  a  series  of  numbers  to  be  differenced  in  the  way  already 
shown,  and  the  various  differences  to  be  designated  as  in  the  follow- 
ing scheme,  which  is  supposed  to  be  a  selection  from  a  series  preceding 
and-  folio  wing  it. 

Function.  1st  Diff.    2d  Diff.     3d  Diff.   4th  Diff.  5tn  Diff. 

—  a    /f'  —  2     /f"'  —  3     yfv 

A  -I  A,,       A     _,  J-.j 

—  — 


•    '         " 

, 

3 


„•     A'",  • 


etc.      etc.      etc.      etc.      etc.      etc. 

It  will  be  seen  that  the  lower  indices  are  chosen  so  as  to 
on  which  line  a  difference  of  any  order  falls.  Thus  all  quantities 
with  index  2  are  on  one  horizontal  line,  those  with  index  |-  =  2£  are 
half  a  line  below,  etc.  This  notation  is  a  little  different  from  that 
used  in  algebra,  but  the  change  need  not  cause  any  confusion.  , 

It  is  shown  in  algebra  that  if  n  be  any  index,  we  have 


the  notation  being  changed  as  in  the  preceding  scheme. 

Now  the  fundamental  hypothesis  of  interpolation  is  that  this 
iormula,  which  can  be  demonstrated  only  for  integral  values  of  ^>k 
true  also  for  fractional  values;  that  is,  for  values  of  the  function  u 
between  those  given  in  the  table  or  in  the  above  scheme.  We  there- 
fore suppose  this  formula  to  express  the  value  of  the  function  u  for 
any  value  of  n  between  0  and  1.  r  .  ,  . 

For  values  between  -f  1  and  -f  2  we  have  only  to  increase  the 
indices,  of  u  and  its  differences  by  unity,  thus: 


.  +  etc,, 


and  by  supposing  n  to  increase  from  0  to  1  in  this  formula  we  shall 
have  values  of  u  from  ul  to  wa. 


62  OF  INTERPOLATION. 

Increasing  the  indices  again  —  that  is,  applying  our  general  foi« 
mulae  to  a  row  of  quantities  one  line  lower  —  we  shall  have 


etc. 


, 
The  equation  (a)  is  known  as  Newton's  formula  of  interpolation. 

28.  Transformations  of  the  Formula  of  Interpolation. 

In  the  equation  (a)  and  those  following  it,  the  formula  of  inter- 
polation is  not  in  its  most  convenient  form.  We  shall  therefore 
transform  it  so  that  the  differences  employed  shall  be  symmetrical 
with  respect  to  the  functions  between  which  the  interpolation  is  to 
be  made. 

In  working  these  transformations  we  shall  suppose  the  sixth  and 
following  orders  of  differences  to  be  so  small  as  not  to  affect  the 
result.  These  differences  being  supposed  zero,  any  two  consecutive 
fifth  differences  may  be  supposed  equal. 

First  transformation.  Let  us  first  find  what  the  original  formula 
(a)  will  become  when,  instead  of  using  the  series  of  differences 

J'*,     J"lf     ^'"j,     A^\,     etc., 
we  use 

J'l,     J"0,     J'"i,     J*f    etc. 

To  effect  the  transformation  we  must  find  the  values  of  the  first 
series  of  differences  in  terms  of  the  second,  and  substitute  them  in 
the  formula  (a). 

We  find,  by  the  mode  of  forming  the  differences, 


for  which,  because  we  suppose  the  values  of  JT  to  be  equal,  we  may  put 
#\  =  *\  +  8J»»; 
/}',  =  A\. 
Making  these  substitutions  in  (a),  we  have 

«„=«.  +  «J't  +  *  (*  ~  l)  (A>\  +  J'",) 


«(»-!) (»-*)  ,, 

1.2.3.4.5  ** 


TRANSFORMATIONS  OF  FORMULAE.  63 

Reducing  by  collecting  the  coefficients  of  equal  differences,  we  find 

«„-«  =  »//'*  +  n(n~ll  A"  +  («  +  l)  «(«-*)  A,n. 
**n       **o  —  !l>     t    i         12          oi  123  » 

(»  +  l)n(*-l)(»-2) 
1.2.3.  4 


.v  . 

1.2.3.4.5  **• 

Second  transformation.    Next,  instead  of  the  series  of  this  last 
formula,  (J), 

J't,    J".,    J'"b    ^0,    etc., 
let  us  use 

/J'_t,    J".,    J"'_4,    J\,    etc. 

To  effect  this  transformation  we  substitute  in  (S)  for  d\,  4"i,  etc., 


The  series  (b)  then  changes  into 

*-^^* 


-        .i 
1.2.3.4 


1.2.3.4.5  '-*' 

tV<f  transformation.  Stirling's  formula.  We  effect  a  third 
transformation  by  taking  the  half  sum  of  the  equations  (b)  and  (c), 
Which  gives  us  a  formula  perfectly  symmetrical  with  respect  to  the 
lines  of  differences,  namely, 


*-^4^+^ 

*>'-!)  n(it--l)(n'-4)k.H-^   .    ^    ^ 

1.2.3.4          °^  1.2.3.4.5  2  T  «0.|  W 

which  is  known  as  Stirling9  s  formula  of  interpolation. 
It  will  be  seen  that  we  have  put 

n*  -  I  for  (n  +  1)  (n  -  1), 
w9  -  4  for  (n  +  2)  (w  -  2), 

etc.  etc. 

Fourth  transformation.   In  the  equation  (5),  instead  of  the  series 
of  differences 

J'h    J".,    J'",,    J".,    etc., 
let  us  use 

A'k,    A',    4"',    &      etc. 


64  OF  INTERPOLATION. 

-.     To  effect  this  we  put 

J"0  =  A'\  -  J'"i5 

JiVo      =     JlVi       _    JV^ 

Making  these  substitutions  in  (£),  it  becomes 
*  -          *J'  11  .f"      ^ 


--        ,iv 
1.2.3.4"  '   ' 


-       .v 
1.2.3.4.5 

transformation.    BesseVs  formula.    Let  us  take  half  the 
sum  of  the  equations  (e)  and  (b).    We  then  have 


1.2.3.4.5 


which  is  commonly  known  as  BesseVs  formula  of  interpolation,  and 
which  is  the  one  most  convenient  to  use  in  practice. 
,  In  applying  this  formula  to  find  a  value  of  the  function  inter- 
mediate between  two  given  values,  we  must  always  suppose  ,^  the 
index  0  to  apply  to  the  given  value  next  preceding  that  to  be  found, 
and  the  index  1  to  apply  to  that  next  following.  The  quantity  n 
will  then  be  a  positive  proper  fraction. 

29.  Example  of  interpolation  to  halves.  If  we  increase  the  loga- 
rithms of  30,  31,  etc.,  already  given,  by  unity,  we  shall  have  the 
logarithms  of  300,  310,  320,  etc.  It  is  required  to  find,  by  interpola- 
tion, the  logarithms  of  the  numbers  half  way  between  the  given  ones 
(omitting  the  first  and  last),  namely,  the  logarithms  of  315,  325,  335, 
etc. 

Here,  the  required  quantities  depending  upon  arguments  half  way 
between  the  given  ones,  we  have  n  =  -J,  and  the  values  of  the  Bessel- 
ian  coefficient,  so  far  as  wanted,  are 

n  (n  -  1)  _         !_ 
2  "  8J 


log  (a,  -  5)  =  log  a,  -     - 


TRANSFORMATIONS  OF  FORMULA.  65 

The  subsequent  terms  are  neglected,  being  insensible.  So,  if  we 
put  a0  and  al  for  any  consecutive  two  of  the  numbers  300,  310,  etc., 
we  have 


(*) 


where  we  put  A±  for  that  first  difference  between  a0  and  alt 

These  two  formulae  are  two  expressions  for  the  same  quantitj 
because  a0  -f-  5  =  al  —  5.     They  are  both  used  in  such  a  way  as  to 
provide  a  check  upon  the  accuracy  of  the  work.  For  this  purpose  we 
compute  the  two  quantities 

log  (a.  +  5)  -  log  a0  =  -^A\  -  -  — •— 1,  1 

1  A"        A"     f  W 

log  a,  -  log  («..+  5)  =  -J'i  +  -  — 1± 1.  J 

The  most  convenient  and  expeditious  way  of  doing  the  work  is 
shown  in  the  accompanying  table,  where  we  give  every  figure  which 
it  is  necessary  to  write,  besides  those  found  on  p.  57.  The  following 
is  the  plan  of  computation: 

Ko.        Log.        Difl.        ^',    *'"•  +  '"'.   £^+£X 


310 

2.49136 

315 

.498  31 

RS4. 

+  689.5 

-  5.5 

-44 

320 

.50515 

UOTt 

325 

.51188 

aao 

668.0 

-  5.1 

-  41 

330 

.51851 

DOO 

r*  Ef  o 

335 

.52504 

bOo 

/?  A  4 

648.5 

-4.8 

-  38 

340 

.53148 

b44 

AQ/f 

345 

.53782 

OO4: 

629.5 

-  4.6 

-  37 

350 

.54407 

/1-f  /> 

355 
360 

.55023 
2.55630 

616 
607 

+  611.5 

-  4.3 

-34 

We  compute  the  right-hand  column  by  the  formula 


using  the  values  of  A  given  in  the  scheme,  p.  57. 

This  mode  of  computing  the  half  sum  of  two  numbers  which  are 
nearly  equal  is  easier  than  adding  and  dividing  by  2. 

In  the  next  two  columns  to  the  left,  the  sixth  place  of  decimals 


66  OF  INTERPOLATION. 

is  added  in  order  that  the  errors  may  not  accumulate  by  the  addition 
of  several  quantities.  This  precaution  should  always  be  taken  when 
the  interpolated  quantities  are  required  to  be  as  accurate  as  the  given 
ones. 

The  fourth  column  from  the  right  is  formed  by  adding  and  sub- 
tracting the  numbers  of  the  second  and  third  columns  according  to 
the  formula  (k).  The  additional  figure  is  now  dropped,  because  no 
longer  necessary  for  accuracy.  The  numbers  thus  formed  are  the 
first  differences  of  the  series  of  logarithms  found  by  inserting  the 
interpolated  logarithms  between  the  given  ones,  as  will  be  seen  by 
equation  (&). 

We  write  the  first  logarithm  of  the  series,  namely, 

log  310  =  2.49136, 

and  then  form  the  subsequent  ones  by  continual  addition  of  the  dif- 
ferences, thus: 

log  315  =  log  310  +  695; 
log  320  =  log  315  +  684; 
log  325  =  log  320  +  673; 
etc.        etc.        etc. 

If  the  work  is  correct,  the  alternate  logarithms  will  agree  with  the 
given  ones  in  the  former  table. 

The  continuance  of  the  above  process  for  a  few  more  numbers, 
say  up  to  450,  is  recommended  to  the  student  as  an  exercise. 

3O.  Interpolation  to  thirds.  Let  us  suppose  the  value  of  a 
quantity  to  be  given  for  every  third  day,  and  the  value  for  every 
day  to  be  required.  By  putting  n  =  -j-  and  applying  formula  (/)  to 
each  successive  given  quantity,  we  shall  have  the  value  for  each  day 
following  one  of  those  given,  and  by  putting  n  =  }  we  shall  have 
values  for  the  second  day  following,  which  will  complete  the  series . 
But  the  interpolation  can  be  executed  by  a  much  more  expeditious 
process,  which  consists  in  computing  the  middle  difference  of  the 
interpolated  quantities  and  finding  the  intermediate  differences  by  a 
secondary  interpolation. 

Let  us  put 

/„  /„  /f,  etc.,  the  given  series  of  quantities; 

/»  /u  f*>  /»>  /4>  etc->  tne  required  interpolated  series; 

A't  A",  etc.,  the  first  differences,  second  differences,  etc.,  of  the 
given  series; 

$',  £",  etc.,  the  first  differences,  second  differences,  etc.,  of  the 
interpolated  series. 


TRANSFORMATIONS  OF  FORMULAS.  67 

We  may  then  put 

/,—/,  =  ^'*      (in  the  given  series); 

/,-/.-*'») 

/,—/!  =  ^  'f  >   (in  the  interpolated  series). 

/.-/.=  *Y) 
We  shall  then  have 

<*'*  +  <*',  +  f|  =  4'*. 

The  value  of  /i  —  /0  =  b\  is  given  by  putting  n  =  $  in  the  Bes- 
selian  formula  (/).     Thus  we  find 

,,       1  1  J".+  J".  ,     1     .,„ 

**  =  3J*~9          2         +i62J    » 

^^0  +  ^_J_ 
r  243  2  1458 

Putting  w  =  |,  we  have  the  value  of/,  ~/0,  that  is,  of 
Thus  we  find 


5     ^t+^.         1 
r  2  r!458 


Subtracting  these  expressions,  we  have 

^=3L^-^'"* 

which  is  most  easily  computed  in  the  form 


We  see  that  the  computation  of  tf'f,  the  middle  difference  of  the 
interpolated  quantities,  is  much  simpler  than  that  of  <?V  It  will 
therefore  facilitate  the  work  to  compute  only  these  middle  differ- 
ences, and  to  find  the  others  by  interpolation. 

This  process  is  again  facilitated,  in  case  the  second  differences  are 
considerable,  by  first  computing  the  second  differences  of  the  inter- 
polated series  on  the  same  plan.  The  formulae  for  this  purpose  are 
derived  as  follows: 

Let  us  put 

<*'!  =/,  -/,- 

The  second  difference  of  which  we  desire  the  value  is  then 

<*»,=  «?'{  -  ff¥ 

The  value  of  S\  is  given  by  the  equation 

tf'j^'j-OJ'j  +  <?',), 


OS  LOGARITHMIC  TABLES. 

and  the  value  of  d'$  is  found  from  that  of  d'±  by  simply  increasing,  the 
indices  of  the  differences  by  unity,  because  it  belongs  to  the  next 
lower  line. 

We  thus  find 


I 
^ 


243          a  1458 


5     A\  +  4\        1 
243          2  1458 

Then  by  subtraction, 


4-       (^-^ 

__        .I._  _1_          _ 
r  243  2  1458  ^     f  W* 

Eeducing  the  first  of  these  terms,  we  have 

^Y-^V;M", 

For  the  second  term, 


whence 

^".  +  ^"a  =  ^^  +  A'"\  -  A'"\  =  2^",  +  A\, 
and 

^".  +  aj"  +  ^">  =  2J»  +  L  j.. 

2  >  ^  2       '* 

For  the  third  term, 

j'"f  _  A'"k  =  J1^. 

For  the  fourth  term,  dropping  the  terms  in  d*  as  too  small  iu 
practice,  we  may  put 

j".  +  aj".  +  ^'%  = 

«g 

The  difference  of  the  fifth  terms  may  also  be  dropped,  because 
they  contain  only  sixth  differences. 

Making  these  substitutions  in  the  value  of  #"3,  we  find 


OF  INTERPOLATION.  69> 

By  this  formula  we  may  compute  every  third  value  of  #",  and 
then  interpolate  the  intermediate  values.  By  means  of  these  values, 
we  find  by  addition  the  intermediate  values  of  d',  of  which  every 
third  value  has  been  computed  by  formula  (m).  Then,  by  continu- 
ally adding  the  values  of  6',  we  find  those  of  the  function/. 

As  an  example  of  the  work,  we  give  the  following  values  of  the. 
sun's  declination  for  every  third  day  of  part  of  July,  1886,  for  Green* 
wich  mean  noon: 

Date.  Q'sDec.  A'  A"          A'" 

1886  o      /         //  in  n  it 


6..  . 

...22 

41 

9 

•> 

—   16 

28. 

3 

—  212. 

4 

9..  . 

...22 

8. 

5 

—  20 

0. 

7 

—  207. 

9 

+  4.5 

12..  . 

...21 

57 

39 

9 

—  23 

28. 

6 

—  203. 

4 

-f  4.5 

15..  . 

...21 

30 

47, 

9 

—  26 

52. 

0 

—  197. 

7 

+  5.7 

18.. 

21 

0 

38. 

2 

—  30 

9. 

7 

The  values  of  Jiv  are  too  small  to  hav-e  any  influence. 

The  whole  work  of  interpolation  is  shown  in  the  following  table> 
fhere,  as  before,  the  right-hand  column  is  that  first  computed,  and 
gives  the  value  of  A'  —  -faA'"  according  to  formula  (m)  : 


Date.  o'sDec.  <?'  d"         A'  - 

1886.  °       '          "  '         "  " 

July  6  ......  22  41  9.2        R  ,  ,  RR  -  23.60 

7  ......  22   34  52.4  '  '  *   *™£  -  23.43        20     Q  ,„ 

8  ......  22   28  12.1  ;  •  «  ^'?*  -  23.27  "            °'87 

9  ......  22  21  8.5  "I  ft£22  -  23-10 

10  ......  22   13   41.9   "  ]   *£™  -  22.93        ^  2g  77 

11  ......  22     5    52.3   "          *5J  -  22.78        ^  ^<77 

12  ......  21  57  39.9  "        g-£  -  22.61 

13  ......  2149     4.9  "  I   **•*»  -  22.42        2  - 

14  ......  21   40     7.5   '  I  ••«§  -  22.  19  "  26  52'21 

15  ......  21  30  47.9  "         19M  -  21.97 

To  make  the  process  in  the  example  clear,  the  computed  differ- 
ences, etc.,  are  printed  in  heavier  type  than  the  interpolated  ones. 

It  is  also  to  be  remarked  that  the  sum  of  the  three  consecutive 

values  of  d",  formed  of  one  computed  value  and  the  interpolated 

values  next  above  and  below  it,  should  be  equal  to  the  difference 

between  the  corresponding  computed  first  differences.    For  instance, 

23".27  +  23".10  +  22".93  =  7'  49".59  -  6'  40".29. 

But  in  the  first  computation  this  condition  will  seldom  be  exactly 
fulfilled,  owing  to  the  errors  arising  from  omitted  decimals  and  other 
sources.  If  the  given  quantities  are  accurate,  the  errors  should  never 


70  LOGARITHMIC  TABLES. 

exceed  half  a  unit  of  the  last  decimal  in  the  given  quantities,  or  five 
units  in  the  additional  decimal  added  on  in  dividing. 

To  correct  these  little  imperfections  after  the  interpolation  of  the 
second  differences,  but  before  that  of  the  first  differences,  the  sum  of 
the  last  two  figures  in  each  triplet  of  second  differences  should  be 
formed,  and  if  it  does  not  agree  with  the  difference  of  the  first  differ- 
ences, the  last  figures  of  the  second  difference  should  each  be  slightly 
altered,  to  make  the  sum  exact. 

The  first  differences  can  then  be  formed  by  addition. 

In  the  same  way,  the  sum  of  three  consecutive  first  differences 
should  be  equal  to  the  difference  between  the  given  quantities.  If, 
as  is  generally  the  case,  this  condition  is  not  exactly  fulfilled,  the 
differences  should  be  altered  accordingly.  This  alteration  may,  how- 
ever, be  made  mentally  while  adding  to  form  the  required  inter- 
polated functions. 

As  an  exercise  for  the  student  we  give  the  continuance  of  the 
sun's  declination  for  the  remainder  of  the  month,  to  be  interpolated 
for  the  intermediate  dates  from  July  15th  onward: 

o       i         n 

July  21 20  27  16.5 

24 19  50  49.1 

27 19  11  22.7 

30 18  29  4.8 

Aug.  2 17  44    3.1 

As  another  exercise  the  logarithms  of  the  intermediate  numbers 
from  998  to  1014  may  be  interpolated  by  the  following  table: 
Number.  Logarithm. 

994 2. 997  386  4 

997 2.9986952 

1000 3.0000000 

1003 3.001 3009 

1006 3.002  598  0 

1009 3. 003  891 2 

1012 3.005  180  5 

1015 3. 006  466  0 

1018 3.007  747  8 

32.  Interpolation  to  fifths.  Let  us  next  investigate  the  formulas 
when  every  fifth  quantity  is  given  and  the  intermediate  ones  are  to 
be  found  by  interpolation.  By  putting  n  =  £  in  the  Besselian  for- 
mula, we  shall  have  the  value  of  the  interpolation  function  second 


OF  INTERPOLATION.  71 

following  one  of  the  given  ones,  and  by  putting  n  =  f  that  third 
following.  The  difference  will  be  the  middle  interpolated  first  dif- 
ference of  the  interpolated  series.  Putting  n  =  £  in  (/  ),  we  have 

2   .,         2.3  J"0  +  J",         2.3.1    .,„ 

m  =  *c  +  r  J'*  -  ^  —  ^  -  +  gr^'"* 

2.3.7.8  A\  +  ^  _    2.3.7.8.1 
^2.3.4.5*  2  2a.3.4.5.56     ** 

Putting  n  =  $,  we  have 

,    3    .          2.3  J"0  +  A'\        2.3.1    .„, 
'  —  "' 


5 


2.3.7.8  ^1Y0  +  ^lv,        8.3.2.7.1    . 
h2.3.4.54  2  i"2'.3.4.5.5' 


The  difference  of  these  expressions,  being  reduced,  gives 
w§  -  u\  =     J'*  -  125^'"*  +  15625  JV* 


The  term  in  ^y  will  not  produce  any  effect  unless  the  fifth  differ- 
ences are  considerable,  and  then  we  may  nearly  always,  in  practice, 
put  $  instead  of  -Jfa. 

The  interpolated  second  differences  opposite  the  given  functions 
are  most  readily  obtained  by  Stirling's  formula,  (d).  Putting  n  =  \> 
we  have  the  following  value  of  the  interpolated  first  differences  im- 
mediately following  a  given  value  of  the  function: 


2  50      •      6.5.25  2 

24 


Again,  putting  n  =  —  -fr,  and  changing  the  signs,  we  find  for  the 
first  difference  next  preceding  a  given  function 


50      •      6.5.25 
24 


6.5.20  »  - 

The  difference  of  these  quantities  gives  the  required  second  dif- 
ference, which  we  find  to  be 


72  LOGARITHMIC  TABLES. 

As  an  example  and  exercise  we  show  the  interpolation  of  loga* 
rithms  when  every  fifth  logarithm  is  given: 


Number. 

Logarithm. 

6' 

3" 

Ap           A" 

1000 

3.0000000 

+  21  661 

1005 

1006 

3.002  166  1 

.0025980 

4319.2 

A  0-1   A     Q 

-4.32 

-  4.31 

-  108 

1007 
1008 
1009 
1010 

.003  029  5 
.0034606 
.0038912 
3.0043214 

'ioi^b.  y 
4310.6 

4306.3 
4302.0 

A  CIA  W    tV 

-  4.30 
-  4.30 
-  4.29 

-  4.28 

+  21  553 
-  107 

1011 

.0047512 

4297.7 

A  Cir\f*i      f 

-  4.27 

1012 
.1013 
1014 
1015 

.0051805 
.0056094 
.006  037  9 
3.0064660 

4293.5 
4289.2 

4285.0 
4280.8 

-  4.26 
-  4.23 
-  4.20 
-4.16 

+  21  446 
+  21  342  ~"  104 

1020 

3.0086002 

1025 

3.0107239 

1030 

3.0128372 

1035 

3.0149403 

1040 

3.0170333 

FORMULAE 


»OE  THE  SOLUTION  OF 


PLANE  AND  SPHERICAL  TRIANGLES. 


REMARKS. 


1.  It  is  better  to  determine  an  angle  by  its  tangent  than  by  its 
sine  or  cosine,  because  a  small  angle  or  an  angle  near  180°  cannot  be 
accurately  determined  by  its  cosine,  nor  one  near  either  90°  or  270° 
by  its  sine, 

Sometimes,  however,  the  data  of  the  problem  are  such  that  the 
angle  can.  be  determined  only  through  its  sine  or  cosine.  Any  un- 
certainty which  may  then  arise  from  the  source  pointed  out  is  then 
inherent  in  the  problem;  e.g.,  if  the  hypothenuse  and  one  side  of  a 
right  triangle  are  0.39808  and  0.39806  respectively  (sixth  and  follow- 
ing decimals  being  omitted),  the  value  of  the  included  angle  may  be 
anywhere  between  0°  25'  and  0°  42',  no  matter  what  method  of  com- 
putation be  adopted. 

2.  If  the  sine  and  cosine  can  be  independently  computed,  their 
agreement  as  to  the  angle  will  generally  serve  as  a  check  on  the 
accuracy  of  the  computation.     If  they  agree,  their  quotient  will  give 
the  tangent. 

3.  It  is  desirable,  when  possible,  to  have  a  check  upon  the  accu- 
racy of  the  computation;  that  is,  to  make  a  computation  which  must 
give  a  certain  result  if  the  work  is  right.     But  no  check  can  give  a 
positive  assurance  of  accuracy:  all  it  can  do  is  to  make  it  more  or 
less  improbable  that  a  mistake  exceeding  a  certain  limit  exists. 

4.  In  the  following  list  several  formulae  are  sometimes  given  as 
applicable  to  the  same  problem.    In  such  cases,  the  most  convenient 
for  the  special  purpose  must  be  chosen. 


PLANE  TRIAXGLE8. 


Notation,     a,  b,  and  c  are  the  three  sides. 

A,  B,  and  C  are  the  opposite  angles. 

PLANE  TRIANGLES. 


Given. 

Required. 

s      a+ 

a,  b,  c, 

^, 

the  three 

one  angle. 

tin  A-  -4  —~  \/ 

sides. 

o 

s(s  —  a) 

A,  B,  C, 

TT         A/(S  ~  a)  (S  —  0)(S  —  C)m 

all  the 

s                ' 

angles. 

tan^  =  -*- 

tan  45-      H 

s  —  V 

tan  -i  G  — 

tCUl    T    ^     —                           « 

s  —  <? 

Checks:  A  +  B  +  C  =  180°; 

a            b            c 

—  —  — 

^  ____ 

sin  A       sin  .5      sin  (7 

b,  c,A, 

B  and  <7, 

7)  /» 

rwo  sides 

the  other 

tan  ItCR        r\  —               ont  4-  A  • 

tu/JJ   ^  ^  jj    -  -    ^  1    —                        ^u  i,  -j  ^n  - 

0    ~T~    C 

and  the 
included 

angles. 

!(£+  C)  =  90°  -|^; 

angle. 

t7=|(^+(7)I|(^Z  (7).5 

Check,  as  before. 

a,  B,  C, 

a  sin  -J  (B  —  C  )  =  (b  —  c)  cos  -J  A  ; 

the 

rt  cos  -£  (B  —  C)  =  (b  -\-  c)  sin  \A. 

remaining 

Having  found  a  and  £  (J?  —  (7),  proceed 

parts. 

as  in  the  last  case. 

o,  5,  A, 

two  sides 

c,  B,  C, 
the  re- 

sin B  =  —  sin  A;  (two  values  of  B.) 

and  the 

maining 

/T       _    -i  QA°          /  A     1      Z?\  • 

O  =  loU    —  {A.  -f-  x>j; 

angle  oppo- 

parts. 

__  £  sin  C'      «  sin  C 

site  one  of 

sin  5  "      sin  ^4  " 

them. 

76 


RIGHT  SPHERICAL  TRIANGLES. 


Given. 

Required. 

a,  A,  B, 

a,  c,  o, 

C  =  180°  -  (A  +  5); 

one  side 

the  re- 

, _  a  sin  ^ 

and  any 

maining 

sin  J.  ' 

two  angles. 

parts. 

a  sin  (7      a  sin  (^4  -f-  J9) 

sin  A                sin  ^4 

RIGHT  SPHERICAL  TRIANGLES. 

c  is  the  hypothenuse. 

«,  b, 

A,  B,  or  c. 

cot  A  =  cot  a  sin  #; 

the  sides 

cot  B  =  cot  J  sin  a\ 

containing 

cos  c  =  cos  a  cos  5; 

the  right 

sin  # 

CJl  f^i     /»     —  — 

angle. 

bill  t/    —       .          -  • 

Bin  -4 

A  and  c. 

sin  c  sin  A  =  sin  a; 

sin  c  cos  ^4  =  cos  a  sin  5; 

cos  c  =  cos  a  cos  5* 

sin  c  sin  ^  =  sin  J; 

B  and  c 

sin  c  cos  B  =  sin  a  cos  5. 

a,  c, 

A,  B}  or  b. 

.        sin  G^ 

sin  A  —    .      -, 

one  side 

sin  c 

and  the  hy- 

cos  .Z?  =  tan  a  cot  C| 

pothenuse. 

^       cos  c 

C/wb    C/     ~—                          • 

cos  a 

a,  A, 

b,  c,  or  B. 

sin  £  =  tan  a  cot  ^4; 

one  side 

sin  a 

and  the 

sin  c  =  —  —  rj 
sin  -4 

opposite 

.              cos  A    ' 

r*i  >i      A.     — 

angle. 

bill   X>     • 

*,&, 

b,  c,  or  A. 

tan  b  =  sin  «  tan  B; 

one  side 

tan  a 

and  the 

tan  c  =  •  ^; 
cos  2r 

adjacent 

cos  ^4  =  cos  a  sin  .#. 

angle. 

c  and  A. 

sin  A  sin  c  =  sin  a; 

sin  ^4  cos  c  =  cos  a  cos  5; 

cos  A  =•  cos  a  sin  B. 

QUADRANTAL  SPHERICAL  TRIANGLES. 


77 


i*iven. 

Required. 

0,  B. 

b  and  A. 

sin  .4  sin  b  =  sin  a  sin  B; 

sin  -4  cos  #  =  cos  B. 

c,A, 

a,  b,  or  B. 

sin  a  =  sin  c  sin  ^4; 

the  hypo- 

tan  b  =  tan  c  cos  A; 

thenuse 

cot  j5  =  cos  c  tan  A 

and  one 

angle. 

a  and  B. 

cos  a  sin  j5  =  cos  A; 

cos  a  cos  J?  =  sin  A  cos  c; 

sin  a  =  sin  -4  sin  c. 

a  and  b. 

cos  a  sin  b  =  cos  -4  sin  c\ 

cos  a  cos  #  =  cos  c. 

A,B, 

a,  b,  or  c. 

cos  .4 

the  two 

sin  ^J 

angles. 

,       cos  jB 
cos  b  =  -.  —  -.  ; 

sm  ^1 

cos  c  =  cot  ^4  cot  B. 

QUADEANTAL  SPHERICAL  TEIANGLES. 


a,  5, 

the  two 

sides. 


a,  (7, 
one  side 
and  the 

angle  oppo- 
site the 

right  side. 


A,  B,  or  C, 

either 
angle. 


A,  B,  or  b. 


A  and  b. 


A  and  B. 


cos  A  = 


c  is  the  omitted  side  equal  to  90°. 
C  is  the  angle  opposite  this  side. 

cos  at 

sin  b' 


cos 


„ 

cos  B  = 

sin  a 

cos  C  =  —  cot 


cot 


sin  -4  =  sin  a  sin  (7; 
tan  B  =  —  cos  a  tan  (7; 
cot  #  =  —  tan  a  cos  (7. 

cos  ^4  sin  Z»  =  cos  a; 
cos  A  cos  b  =  —  sin  a  cos  C. 
sin  -4.  =  sin  a  sin  & 

cos  A  sin  ^  =  cos  a  sin  (7; 
cos  ^4  cos  B  =  —  cos  C. 


78 


QUADRANTAL  SPHERICAL  TRIANGLES. 


Given. 

one  angle 

and  the 

adjacent 

side. 


0,  A, 
one  side 
and  the 
opposite 

angle. 


one  angle 
and  the 
angle  oppo- 
site the 
right  side. 

A,B, 
two  angles. 


Required. 
a,  B,  or  (7. 


a  and  B. 


a  and  C. 


b,  B,  or  C. 


a,  b,  or  B. 


a,  b9  or  C. 


a  and  C. 


I  and  a 


cos  a  =  cos  A  sin  5; 
tan  B  =  sin  A  tan  b; 
cot  C  =  —  cot  A  cos  £. 

sin  a  sin  B  =  sin  J.  sin  J; 
sin  a  cos  .Z?  =  cos  &; 

cos  a  =  cos  A  sin  £. 

sin  a  sin  (7  =  sin  A  \ 

sin  a  cos  C  =  —  cos  J.  cos 


5  = 


cos  a 


cos  A' 
sin  .#  =  cot  a  tan  .4; 

.     ~      sin  A 

sm  #  =  — . 

sin  a 


sin  A 

sm  a%=  - — ^ 
sm  C" 

cos  b  =  —  tan  ^4  cot  C"; 

D           cos  (7 
cosjS  = 3. 


cot  a  =  cot  A  sin  B; 
cot  #  =  sin  A  cot  .5; 
cos  (7  =  —  cos  A  cos  #. 

sin  £7  sin  a  —  sin  ^4; 
sin  (7  cos  a  =  cos  ^4  sin  5; 
cos  (7  =  —  cos  A  cos  5. 

sin  C  sin  5  =  sin  B\ 

sin  (7  cos  b  =  sin  ^4  cos  B. 


SPHERICAL  TRIANGLES  IN  GENERAL. 


SPHERICAL  TRIANGLES  IN  GENERAL. 


Given. 

Required. 

a,  b,  c, 

A,  B,  G, 

-     \                  ), 

the  three 
sides. 

the  three 
angles. 

rr           A/Sm  (S~~a)  SI*1  (S~#)  Sm  (5  —  C) 

sins 

tan  4  A                           • 

sin  (s  —  a) 

fin  47?                                   • 

tail  TT  -O    —   —  :  7T> 

sm  (5  —  b)' 

tnr»  4  ^  — 

sm  (s  —  £) 

p,     ,     sin  a      sin  5      sin  c 

sin  J[       sin  B      sin  (7* 

a,  5,  C9 

A  and  <?, 

sin  c  sin  A  =  sin  a  sin  C; 

two  sides 

one  angle 

sin  c  cos  A  =  cos  a  sin  b  —  sin  «  cos  b  cos  (7; 

and  the 

and  the 

cos  c  =  cos  a  cos  5  +  sin  a  sin  5  cos  C. 

included 

remaining 

angle. 

side. 

B  and  c. 

sin  c  sin  B  =  sin  5  sin  C; 

sin  c  cos  .Z?  =  sin  a  cos  5  —  cos  a  sin  5  cos  C. 

If    addition  and  subtraction  logarithms 

are  not  available  for  this  computation,  we 

may  compute  Jc  and  K  from 

Tc  sin  K  =  sin  a  cos  (7; 

&  cos  K  =  cos  a. 

Then 

sin  c  cos  -4  =;  Jc  sin  (#  —  JT); 

cos  c  =  k  cos  (b  —  JT). 

Also, 

A  sin  H  =  sin  #  cos  C\ 

h  cos  If  =  cos  J. 

Then 

sin  c  cos  ^  =  h  sin  (#  —  If); 

cos  c  =  h  cos  (a  —  5"). 

A,  B,  c, 

sin  -J  c  sin  -J  (  J.  —  .Z?)  =  cos  %  Csin  %  (a  —  b)  ; 

all  the 

sin  I  c  cos  |  (^4  —  ^)  =  sin-J-  C7sinJ(«  +  £); 

remaining 

cos|csin-£  (A-\-B)=  cos-J  (7cos|  (a  —  b); 

pasts. 

cosij  ccos-J  (-44-2?)  =  sin  J  C  cos  J  (a  -\-b). 

80 


SPHERICAL   TRIANGLES. 


Given. 

Required. 

a,  I,  A, 

two  sides 

B,  C,  c, 

all  the 

.     D       sin  A  sin  b         ,  , 
sm  B  =  —  :  (two  values  of  £); 
sin  CL 

and  an 

remaining 

4-0^,  i  n       cos  i(a  —  ^)  c°t  %(A  -\-  B) 

opposite 

parts. 

cos  %  (a  +  b) 

angle. 

cos  4  (A  4-  i?)  tan  !(«  +  &) 
tan  *  c  —                                   <s  \      i     ^ 

A,  B,  c, 

a  and  C, 

sin  C  sin  a  =  sin  A  sin  c; 

two  angles 

one  side 

sin  (7  cos  a  =  cos  ^4  sin  B  -\-  sin  J.  cos  B  cos  0; 

and  the 

and  the 

cos  (7  =  —  cos  A  cos  .5-f-sin  A  sin^B  cos  c. 

included 

third  angle. 

side. 

1  and  C. 

sin  Csmb  =  sin  .#  sin  c\ 

sin  (7  cos  5  =  sin  A  cos  ^  -f-  cos  A  sin  J5  cos  c. 

If  we  compute  &  and  K  from 

&  sin  K  =  cos  ^4, 

&  cos  -5T  =  sin  ^4.  cos  c, 

then          sin  (7  cos  a  =  k  cos  (^  —  K)\ 

cos  (7  =  k  sin  (#  —  -ZT). 

If  we  compute  h  and  H  from 

A  sin  ^T  =  cos  By 

h  cos  If  =  sin  j9  cos  c, 

;hen            sin  (7  cos  £  =  h  cos  (^4  —  #); 

- 

cos  C  —  h  sin  (A  —  H). 

a,  1,  C, 

sin  £  (7sin£(a-|-  Z>)  =  sin£c  cosj(^4  —  B); 

all  the 

sin  £  tfcos  I  (a  -f  Z>)  =  cos  |  c  cos  J  (^4  +  B); 

remaining 

cos^(7sinj(«  —  b)  =  sin£c  sin  %(A  —  B); 

parts. 

cos  -J-  C'cos  J  («  —  b)  =  cos  ^  c  sin  £  (^4  +  £). 

A,  B,  a, 

two  angles 

&,  c,  C, 
all  the 

.     ..       sin  a  sin  B 

en  T|    /)    —                                                             /-f-TTT/^    iTrtli-i/%ci    r\-r    A\* 

Dill  U    —    :  1  LWU    YdtlUUo  \JL  U}* 

sin  A 

and  an 
opposite 

remaining 
parts. 

^*-   _  cos  1  (^  +  B)  tan  i  (a  +  b)m 

wall  -g-  (/   —    '  "                           1  /  /f      "      7?\                    ' 

side. 

cos  -|-  (^  —  ^)  cot  -|-  (^4  -j-  ^) 

COS  %  (a  +  ^) 

A,  B,  (7, 

a,  I,  c, 

#=£(^  +  £+(7); 

the  three 

the  three 

P-4/                     -cos-S' 

angles. 

sides. 

K  cos(^-^)  cos  (#-.#)  cos  (#-(7)' 

tan  Ja  =  P  cos  ($  —  ^4); 

tan  %b  =  P  cos  ($  —  ^)^ 

tan^c  =  Pcos  (S  —  C). 

TABLES. 


TABLE   I. 

COMMON  LOGARITHMS 

OF   NUMBERS. 

X. 

Log. 

N. 

Log. 

N. 

Log. 

N. 

Log. 

N. 

Log. 

0 

1 

2 
3 

—  Infinity. 

30 

31 
32 
33 

I.477I2 

60 

61 
62 
63 

1.77815 

90 

91 
92 
93 

1.95424 

120 

121 
122 
123 

2.07  918 

O.OO  OOO 
0.30  103 

0.47  712 

1.49  136 
I.505I5 
1.51  851 

1.78533 
1.79239 
1.79934 

.95904 
.96  379 
.96  848 

2.08  2>9 
2.08636 

2.08  991 

4 
5 
6 

0.60  206 
0.69  897 

0.77815 

34 
35 
36 

I.53I43 
1.54407 
1.55630 

64 
65 
66 

i.  80618 
1.81  291 
1.81  954 

94 
95 
96 

•97  313 

.97  772 
.98  227 

124 
125 
126 

2.09342 

2.09691 
2.10037 

7 
8 
9 

10 

11 

12 
13 

0.84  510 
.0.90  309 
0.95  424 

37 
38 
39 

40 

41 
42 
43 

1  .  56  820 
1.57978 
1.59  106 

67 

68 
69 

70 

71 

72 
73 

1.82607 
1.83251 
1.83  885 

97 
98 
99 

100 

101 

102 
103 

.98  677 
.99  123 
•99  564 

127 

128 
129 

130 

131 
132 
133 

2.10  380 
2.IO  721 
2.  1  1  059 

I.OOOOO 

1.  60  206 

1.84  510 

2.00000 

2.  ii  394 

.04  139 

.07  918 
.11  394 

1.61  278 
1.62323 
1.63347 

1.85  126 

1.85733 
1.86332 

2.00432 
2.00  860 
2.01  284 

2.  1  1  727 
2.12057 
2.12385 

14 
15 
16 

.14613 
.17609 
.20412 

44 
45 
46 

1.64345 
1.65321 
1.66  276 

74 
75 
76 

1.86923 
1.87  506 
i.  88081 

104 
105 
106 

2.01  703 
2.02  119 
2.02  531 

134 
135 
136 

2.12  710 
2,13033 
2.13354 

17 
18 
19 

20 

21 
22 
23 

.23043 
.25  527 
.27  875 

47 
48 
49 

50 

51 
52 
53 

1.67  210 
1.68  124 
1.69  020 

77 
78 
79 

80 

81 

82 
83 

1.88649 
1.89  209 
1^89763 

107 
108 
109 

110 

111 
112 
113 

2.02  938 
2.03  342 
2.03743 

137 
138 
139 

140 

141 
142 
143 

2.13672 
2.13988 
2.I430I 

.30  103 

1.69897 

1.90309 

2.04139 

2.I46I3 

.32  222 
.34242 
.36  173 

1.70757 
1.71  600 
1.72428 

i  .90  849 
1.91  381 
1.91  908 

2.04532 
2.04922 
2.05  308 

2.14  922 
2.15229 
2.15534 

24 
25 
26 

.38021 

•39794 
•41  497 

54 
55 
66 

1.73239 
1.74036 
1.74819 

84 
85 
86 

i  .92  428 
1.92942 
1.93450 

114 
115 
116 

2.O5  600 
2.06070 
2.06446 

144 
145 
146 

2.15836 
2.16  137 
2.16435 

27 
28 
29 

30 

.43  136 
.44716 
.46  240 

57 
58 
59 

60 

1.75587 
I  76  343 
1.77085 

87 
88 
89 

90 

1.93952 

1.94448 
1-94939 

117 
118 
119 

120 

2.06  819 

2.07  188 
2.07  555 

147 
148 
149 

150 

2.16732 
2.17  026 
2.17  319 

1.477" 

1.77815 

1.95424 

2.07918 

2.17  609 

TABLE  I. 


N. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

100 

01 
02 
03 

04 
05 
06 

07 
08 
09 

110 

11 
12 
13 

14 
15 
16 

17 
18 
19 

120 

21 
22 
23 

24 
25 
26 

27 
28 
29 

130 

31 
32 
33 

34 
35 
36 

37 
38 
39 

140 

41 
42 
43 

44 
45 
46 

47 
48 
49 

150 

oo  ooo 

043 

087 

130 

173 

217 

260 

303 

346 

389 

i 

2 

3 
4 
5 
6 

7 
8 

9 

i 

2 

3 

4 
5 
6 
7 
8 

9 

i 

2 

3 
4 
5 
6 
7 
8 

9 

i 

2 

3 
4 
5 
6 

7 
8 

9 

i 

2 

3 
4 
5 
6 

7 
8 

9 

44 

4-4 

8.8 
13  a 
17-6 

22.0 
26-4 

30.8 
35-2 
39-6 

4« 

4' 

8.2 

12.3 

16.4 

20.5 

24.6 
28.7 
32.8 

36-9 

38 

3-8 
7.6 
ix.  4 
15-2 
19.0 

22.8 
26.6 

3°  4 
34  .3 

35 

3-5 
7-o 
10.5 
14.0 
17  5 

21.  0 

«4  5 
28.0 

31-5 

32 

3-2 
6-4 
9-6 

12.8 

16.0 
19.2 

22-4 

25-6 
28.8 

43 

43 

8.6 
12.9 
17.2 

21.  S 

25.8 
30.1 

34  4 
38.7 

40 

4-o 
8.0 

12.0 

16.0 

2O-  0 

24.0 
28.0 
32.0 

36.0 

37 

3-7 

7-4 

IX.  I 

14-8 
18-5 

22.2 
25.9 
29.6 

33-3 

.34 

3-4 
6-8 

10  2 

I3-6 
17.0 

20.4 

23.8 
27.2 

30.6 

31 

31 

6.2 

93 
12.4 

IS  5 
18.6 
21.7 
24-8 
27.9 

49 

4 
8. 

12. 

16. 

21. 

25 
29. 

33-6 
37-8 

39 

39 

7-8 
11.7 
156 
'9  5 
23.4 

27  3 
31-2 
35  « 

36 
3.6 
7-a 
10.8 
14  4 
18  o 

21  6 

25.3 
28.8 

32  4 

33 

33 
6.6 

99 

13  3 

I6'5 
19-8 

23.  x- 

26.4 

29  7 

30 

3-o 
6.0 

9-o 

12.  0 
15-0 

18.0 

21.  0 
24.0 
37-0 

432 
860 

oi  284 

703 

02  119 

531 

938 
03  342 

743 

475 
903 
326 

725 
1  60 

572 

979 
383 
782 

518 

945 
368 

787 

202 

612 

*oi9 

423 
822 

561 
988 
410 

828 

243 
653 
*o6o 

463 
862 

604 
*030 
452 

870 
284 
694 

*IOO 

503 
902 

647 

*072 

494 
912 
325 
735 

*I4I 

543 
941 

689 

*ii5 

536 

953 
366 
776 

*i8i 

583 
981 

732 

*I57 
578 

995 
407 
816 

*222 
623 
*02I 

775 
*i99 
620 

*O36 
449 
857 

*262 
663 

*o6o 

817 

*242 
662 

*078 
490 
898 

*3Q2 
703 
*IOO 

04  139 

179 

218 

258 

•297 

336 

376 

415 

454 

493 

532 
922 
05  308 

690 
06  070 
446 

819 
07  1  88 
555 

52' 
96i 

346 

729 
1  08 
483 

856 
225 
591 

610 

999 

385 

767 

H5 
521 

893 
262 
628 

650 
*038 
423 

805 
183 
558 

930 
298 
664 

689 

*077 
461 

843 

221 

595 
967 

335 
700 

727 
*iiS 
500 

881 
258 
633 

*oo4 
372 
737 

766 
*I54 
538 

918 

296 

670 

*04i 
408 
773 

805 

="192 

576 
956 

333 
707 

*078 

445 
809 

844 

*23I 

614 

994 
371 
744 

*ii5 

482 
846 

883 
*269 
652 

*032 

408 

781 

*I5I 

518 
882 

918 

954 

990 

*027 

*o63 

*099 

*I35 

*i7i 

*2O7 

*243 

08  279 
636 
991 

09  342 
691 
10  037 

380 
721 
n  059 

3H 
672 

*026 

377 
726 
072 

415 
755 
093 

350 
707 
*o6i 

412 
760 
1  06 

449 
789 
126 

386 

743 
*096 

447 
795 
140 

483 
823 
1  60 

422 
778 

*I32 

482 
830 

175 

517 
857 
193 

458 
814 
*i67 

517 
864 
209 

551 

890 
227 

493 
849 

*202 

552 
899 

243 

585 
924 
261 

529 
884 

*237 

587 
934 
278 

619 
958 
294 

565 

920 

*272 

621 
968 
312 

653 

992 

327 

600 
955 
*3<>7 

656 
*oo3 
346 
687 

*025 

36  1- 

394 

428 

461 

494 

528 

561 

594 

628 

661 

694 

727 

12  057 
385 
710 
13033 

354 
672 
988 
14  301 

760 
090 
418 

743 
066 
386 

704 
*oi9 
333 

793 
123 
450 

775 
098 
418 

735 
*o5i 
364 

826 

483 
808 

130 
450 

767 

*082 

395 

860 
189 
516 

840 
162 
481 

799 
*ii4 
426 

893 

222 
548 

872 
194 
513 
830 

*I45 
457 

926 

254 
581 

3 

545 
862 
*i76 
489 

959 
287 

613 

937 
258 
577 

893 

*208 

520 

992 
320 
646 

969 
290 
609 

925 
*239 
55i 

*O24 

$ 

*OOI 

322 
640 

956 

*27O 

582 

613 

644 

675 

706 

737 

768 

799 

829 

860 

891 

922 
15  229 

534 

836 
16  137 
435 

732 
17  026 

319 

953 
259 

564 

866 
167 
465 

761 
056 

348 

983 
290 

594 

897 
197 
495 
791 
085 
377 

*oi4 
320 
625 

927 
227 

524 

820 
114 
406 

*045 
351 
655 

957 
256 

554 
850 
H3 
435 

*076 
38i 
685 

987 
286 
584 
879 

f? 

464 

*io6 
412 
715 
*oi7 
316 
613 
909 

202 

493 

*i37 
442 
746 

*047 
346 
643 
938 
231 
522 

*i68 

473 
776 

*077 
376 
673 

967 
260 

55i 

*i98 

503 
806 

*io7 
406 
702 

289 
580 

609 
O 

638 

667 
2 

696 
3 

725 
4 

754 

MIBMMMM 

5 

782 

-••«•—•• 

6 

811 
7 

840 

8 

869 
9 

N. 

1 

Prop.  Pts. 

LOGARITHMS  OF  NUMBERS. 


N. 

O 

1 

2 

3 

4 

5 

6 

r 

8 

9 

Prop.  Pts. 

150 

17  609 

638 

667 

696 

725 

754 

782 

811 

840 

869 

51 

898 

926 

955 

984 

*oi3 

"041 

*o7o 

*O99 

*I27 

*i$6 

29 

28 

52 

18  184 

213 

241 

270 

298 

327 

355 

384 

412 

441 

53 

469 

498 

526 

554 

583 

611 

639 

667 

696 

724 

1 
2 

2.9 

5.8 

2.8 
5.6 

54 

752 

780 

808 

837 

865 

893 

921 

949 

977 

*oos 

3 

8.7 

8.4 

55 

19033 

061 

089 

117 

145 

173 

201 

229 

257 

285 

4 

11.6 

11.2 

56 

312 

340 

368 

396 

424 

45  l 

479 

507 

535 

562 

5 

14.5 

14.0 

57 

590 

618 

645 

673 

700 

^728 

*756 

783 

811 

838 

6 

17.4 

16.8 

58 

866 

893 

921 

948 

976 

*o58 

*o85 

*II2 

7 

20.3 

19.6 

59 

20  140 

167 

194 

222 

249 

276 

303 

330 

358 

385 

8 

0 

23.2 

22.4 

OK  0 

160 

412 

439 

466 

493 

520 

548 

575 

602 

629 

656 

<J 

* 

<i«J  ,  £t 

61 

683 

710 

*737 

763 

*79° 

817 

844 

871 

898 

925 

27 

20 

62 

952 

978 

*O32 

*o85 

*II2 

*I39 

*i65 

+192 

1 

o  7 

9  ft 

63 

21  219 

245 

272 

299 

325 

352 

378 

405 

43i 

458 

J. 

2 

&  .  f 
5.4 

4.  V 

5.2 

64 

484 

5" 

537 

564 

590 

617 

643 

669 

696 

722 

3 

8.1 

7.8 

65. 

748 

775 

80  1 

827 

854 

880 

906 

932 

958 

985 

4 

10.8 

10.4 

66' 

22  Oil 

037 

063 

089 

115 

141 

167 

194 

220 

246 

5 

13.5 

13.0 

Ifi  9 

1C  £» 

67 
68 
69 

272 
531 

,  789 

298 

557 
814 

324 
583 
840 

350 
608 
866 

376 
634 
891 

401 
660 
917 

427 

686 
943 

453 
712 
968 

479 
737 
994 

7*3 

7 
8 
9 

ID  .  £ 

18.9 
21.6 
24.3 

10  .  o 
18.2 
20.8 
23.4 

170 

23  045 

070 

096 

121 

147 

172 

198 

223 

249 

274 

71 

300 

325 

350 

376 

401 

426 

452 

477 

502 

528 

25 

72 

553 

603 

629 

654 

679 

704 

729 

754 

779 

1  2.5 

73 

805 

830 

855 

880 

905 

930 

955 

980 

*<x>5 

*030 

2  5.0 

74 

24  055 

080 

105 

130 

155 

180 

204 

229 

254 

279 

3  7.5 

75 

304 

329 

353 

378 

403 

428 

452 

477 

502 

527 

4  10.0 

76 

576 

601 

625 

650 

674 

699 

724 

748 

773 

5  12.5 

77 

797 

822 

846 

871 

895 

920 

944 

969 

993 

*oi8 

6  15.0 
7  17.5 

78 

25  042 

066 

091 

115 

139 

164 

1  88 

212 

237 

261 

8  20.0 

79 

285 

310 

334 

358 

382 

406 

431 

455 

479 

503 

180 

527 

551 

573 

600 

624 

648 

672 

696 

720 

744 

81 

768 

792 

816 

840 

864 

888 

912 

935 

959 

983 

34 

H 

82 

26  007 

031 

055 

079 

102 

126 

150 

174 

198 

221 

1 

2.4 

2.3 

83 

245 

269 

293 

3I6 

340 

364 

387 

411 

435 

458 

2 

4.8 

4.6 

84 

482 

5°5 

529 

553 

576 

600 

623 

647 

670 

694 

3 

7.2 

6.9 

85 

717 

764 

788 

8n 

834 

858 

881 

905 

928 

4 

9.G 

9.2 

86 

951 

975 

998 

*02I 

*045 

*o68 

"091 

"114 

*i38 

*i6i 

5 
6 

12.0 
14.4 

11.5 
13.8 

87 

27  184 

207 

231 

254 

277 

300 

323 

346 

370 

393 

7 

16.8 

16.1 

88 
89 

416 
646 

439 
669 

g 

485 
715 

508 
738 

531 
761 

554 

784 

577 
807 

600 
830 

623 
852 

8 
9 

19.2 
21.6 

18.4 
20.7 

190 

875 

898 

921 

944 

967 

989 

*OI2 

*Q35 

*os8 

*o8i 

99 

91 

91 

28  103 

126 

149 

171 

194 

217 

240 

262 

285 

307 

mm 

21 

92 

330 

353 

398 

421 

443 

466 

488 

533 

1 

2.2 

2.1 

93 

556 

578 

601 

623 

646 

668 

691 

713 

735 

758 

2 

4.4 

4.2  i 

94 

780 

803 

825 

847 

870 

892 

914 

937 

959 

981 

3 

4 

6.6 

80 

6.3 

Q  4  , 

95 
96 

29  003 

226 

026 

248 

048 
270 

070 
292 

092 

336 

137 
358 

58 

ill 

403 

203 

425 

rr 

5 
6 

*  O 

11.0 
13.2 

O  .  rr 

10.5 
12.6 

97 

447 

469 

491 

513 

535 

557 

579 

601 

623 

645 

7 

15.4 

14.7 

98 

667 

688 

710 

732 

754 

776 

798 

820 

842 

863 

8 

17.6 

16.3 

99 

885 

907 

929 

951 

973 

994 

*oi6 

*o38 

*o6o  *o8i 

9 

19.818.9 

200 

30  103 

125 

146 

168 

190 

211 

233 

255 

276  298 

N. 

O 

1 

2 

3 

4 

5 

6 

7 

8  1  9 

Prop.  Pts. 

TABLE  I. 


N. 

O 

1 

9 

3 

4 

5 

6 

7 

8 

O 

Prop. 

Pts.  1 

200 

01 

02 
03 

04 
05 
06 

07 

08 
09 

210 

11 
12 
13 

14 
15 
16 

17 
>18 
19 

220 

21 
22 
23 

24 
25 
26 

27 

28 
29 

280 

31 
32 
33 

34 
35 

36 

37 

38 
39 

240 

41 
42 
43 

44 
45 
46 

47 

48 
49 

250 

•••••MM 

N. 

30  103 

123 

146 

168 

190 

211 

233 

255 

276 

298 

2 

1  2 

2  4 
3  6 
4  8 
511 
613 
7  15 
817 
9  19 

1 

2 
3 

4 
5 
6 
7 
8 
9 

1 
2 
3 
4 
5 
6 
7 
8 
9 

1 

2 
3 

4 
5 
6 
7 
8 
9 

1 
2 
3 
4 
5 
6 
7 
8 
9 

~Pr 

2  21 

.2  2.1 

.4  4.2 
.6  6.3 
.8  8.4 
.0  10.5 
.2  12.6 
.4  14.7 
..6  16.8 
.818.9 

20 

2.0 

4.0 
6.0 
8.0 
10.0 
12.0 
14.0 
16.0 
18.0 

19 

1.9 
3.8 
5.7 
7.6 
9.5 
11.4 
13.3 
15.2 
17.1 

18 

1.8 
3.6 

5.4 
7.2 
9.0 
10.8 
12.6 
14.4 
16.2 

17 

1.7 
3.4 
5.1 

6.8 
8.5 
10.2 
11.9 
13.6 
15.3 

»mmm*m»wiBmmmmv 

op.  Pts. 

320 

535 
750 

963 
3i  175 

387 

597 
32  015 

34i 

557 
771 

984 
197 
408 

618 

827 
035 

363 
578 

792 

*oo6 
218 
429 

sis 

040 
056 

& 

814 

*027 

239 

450 
660 

869 

077 

406 
621 
835 
*048 
260 
471 
681 
890 
098 

428 

643 
856 

*o69 
281 
492 

702 

9ii 
118 

449 
664 
878 

*09i 
302 
513 

723 
93i 
139 

471 
685 

899 

*II2 
323 

534 

744 
952 
1  60 

492 
707 
920 

*I33 
345 
555 

765 

973 
181 

5H 
728 
942 

*I54 
366 
576 

785 
994 
20  1 

222 

243 

263 

284 

305 

325 

346 

366 

387 

408 

428 

634 
838 

33  041 

244 

445 
646 
846 
34  044 

449 
654 
858 

062 
264 
465 

666 
866 
064 

469 

6/5 
879 

082 
284 
486 

686 
885 
084 

490 

695 
899 

102 

304 
506 

706 
90S 
104 

510 

715 
919 

122 

325 
526 

726 

925 

124 

53i 
736 
940 

H3 
345 
546 

746 
945 
U3 

552 
756 
960 

163 

3S 
566 

766 
965 
163 

572 

777 
980 

183 

385 
586 

786 
985 
183 

593 
797 

*OOI 

203 
405 
606 

806 
*oo5 
203 

613 
818 

*02I 

224 
425 
626 

826 

*O25 

223 

242 

262 

282 

301 

321 

34i 

361 

380 

400 

420 

439 
635 
830 

35  025 

218 
411 

603 
793 
984 

459 
655 
850 

044 
238 
430 
622 

813 
*oc>3 

479 
674 
869 

064 
257 
449 
641 
832 

*02I 

498 

083 
276 
468 

660 
*85' 

*O4O 

5l8 

713 
908 

I  O2 

679 
870 

*059 

537 
733 
928 

122 

315 
507 

698 

889 

*078 

557 
753 
947 

141 
334 
526 

717 

908 
*097 

577 
772 

967 
160 
353 
545 

736 
927 
*u6 

596 
792 
986 

1  80 
372 
564 

755 
946 
*i35 

616 

Su 
*oo5 

199 

392 
583 

774 
965 
*I54 

36  173 

192 

211 

229 

248 

267 

286 

305 

324 

342 

361 
549 
736 

922 
37  107 
291 

£S 

840 

380 
568 
754 
940 
125 
310 

493 
676 
858 

773 

959 

144 

328 

694 
876 

418 
605 
791 

977 
162 

346 

530 
712 

894 

436 
624 
810 

996 
181 
365 
548 

73i 
912 

i55 

642 

829 

*oi4 
199 
383 
566 
749 
93i 

474 
661 
847 

*033 

218 
401 

%* 
767 

949 

493 
680 
866 

*o5i 
236 
420 

603 
785 
967 

511 

*070 
254 
438 

621 

803 
985 

530 
717 

903 
*o88 
273 
457 

639 

822 
*oo3 

38  021 

039 

057 

075 

093 

112 

130 

148 

166 

184 

2O2 
382 
561 

739 
917 

39<>94 
270 

445 
620 

220 

399 
578 

757 
934 
in 

287 
463 
637 

238 
417 
596 

775 
952 
129 

305 
480 

655 

256 

I35 
614 

792 
970 
146 

322 
498 
672 

274 

453 
632 

Sio 
987 
164 

340 

I15 
690 

292 

471 
650 

828 

*oo5 
182 

358 
533 
707 

310 
846 

*023 

199 

375 
550 
724 

328 
507 
686 

863 
*04i 
217 

393 

568 
742 

346 
525 
703 
881 
*os8 
235 

410 
585 
759 

364 
543 
721 

899 
*076 
252 

428 
602 

777 

794 

MMMMBBMHI 

0 

Sn 

|T 

829 
2 

846 

^—  —  —  : 

3 

863 
4 

881 

•M 

5 

898 
6 

915 

mammmmm 

7 

933 

8 

950 
O 

LOGARITHMS  OF  NUMBERS. 


N. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Pr< 

>p.  PK 

250 

39  794 

811 

829 

846 

863 

88  1 

898 

915 

933 

950 

MMMMHM 

••MBMMMMMM 

51 

967 

985 

*002 

*io6 

*I23 

18 

52 
53 

40  140 
312 

157 
329 

I7I 
346 

192 

364 

209 

226 
398 

243 
415 

261 
432 

278 
449 

295 
466 

1 

2 

1.8 
3  6 

54 

483 

500 

5I8 

535 

552 

569 

586 

603 

620 

637 

3 

5.4 

55 

654 

671 

688 

705 

722 

739 

756 

773 

790 

8^7 

4 

7.2 

56 

824 

841 

858 

875 

892 

909 

926 

943 

960 

976 

5 

9.0 

57 

993 

*OIO 

*027 

*044 

*o6i 

*07g 

*095 

*m 

*I28 

*i45 

6 

10.8 

58 

m 
41  162 

179 

196 

212 

229 

246 

263 

280 

296 

3^3 

7 

12.6 

59 

330 

347 

363 

380 

397 

414 

430 

447 

464 

481 

8 

14.4 

260 

497 

5H 

531 

547 

564 

581 

597 

614 

631 

647 

' 

61 

664 

68  1 

697 

7H 

731 

747 

764 

780 

797 

814 

17 

62 

830 

847 

863 

880 

896 

913 

*922 

946 

963 

979 

i 

1  7 

63 

996 

*OI2 

*029 

*045 

*062 

*o78 

*m 

"127 

*I44 

J. 

2 

-l.i 

3.4 

64 
65 

42  160 
325 

177 
341 

193 
357 

210 

374 

226 
390 

406 

259 
423 

275 
439 

292 
455 

308 

472 

3 

4 

5.1 
6.8 

66 

488 

504 

521 

537 

553 

570 

586 

602 

619 

635 

5 

8.5 

67 

651 

667 

684 

700 

716 

732 

749 

765 

781 

797 

6 

7 

10.2 

UQ 

68 

8i3 

830 

846 

862 

878 

*894 

*9" 

927 

*943 

959 

t 

.  y 

1  Q  ft 

69 

975 

991 

*oo8 

*034 

*040 

*o88 

*I20 

9 

lo  .  O 

15.3 

270 

43  136 

152 

169 

185 

201 

217 

233 

249 

265 

281 

71 

297 

313 

329 

345 

361 

377 

393 

409 

425 

441 

16 

72 

457 

473 

489 

505 

521 

537 

553 

569 

584 

600 

1 

1.8 

73 

616 

632 

648 

664 

680 

696 

712 

727 

743 

759 

2 

74 

775 

791 

807 

823 

838 

854 

870 

886 

902 

*9'7 

3 

4^8 

75 

933 

949 

965 

981 

996 

*OI2 

*028 

*°44 

*o59 

4 

6.4 

76 

44091 

107 

122 

138 

154 

170 

185 

20  1 

217 

232 

5 

Q 

8.0 
0  (i 

77 

248 

264 

279 

295 

3" 

326 

342 

358 

373 

389 

7 

«/  •  vl 

11.2 

78 
79 

404 
560 

420 
576 

436 
592 

45i 
607 

467 
623 

483 
638 

498 
654 

669 

S2Q 
685 

545 
700 

8 
9 

12.8 
14.4 

280 

716 

73  i 

747 

762 

778 

793 

809 

824 

840 

855 

81 

871 

886 

902 

917 

932 

948 

963 

979 

994 

*OIO 

15 

82 

45  025 

040 

056 

071 

086 

1  02 

117 

133 

148 

163 

1 

1.5 

83 

179 

194 

209 

225 

240 

255 

271 

286 

301 

317 

2 

3.0 

84 

332 

347 

362 

378 

393 

408 

423 

439 

454 

469 

3 

4" 

4.5 
6f\ 

85 

484 

500 

515 

530 

545 

561 

576 

606 

621 

• 

.0 
7e« 

86 

637 

652 

667 

682 

697 

712 

728 

743 

758 

773 

6 

.5 
9.0 

87 

788 

803 

818 

834 

849 

864 

879 

894 

909 

924 

7 

10.5 

88 

939 

954 

969 

984 

*000 

*oi5 

*O3O 

*°45 

*o6o 

"075 

8 

12.0 

89 

46  090 

105 

1  20 

135 

150 

165 

1  80 

195 

210 

225 

9 

13.5 

200 

240 

255 

270 

285 

300 

315 

330 

345 

359 

374 

91 

389 

404 

419 

434 

449 

464 

479 

494 

509 

523 

14 

92 
93 

g 

553 
702 

568 
716 

583 

746 

613 
761 

627 
776 

642 
790 

657 
805 

820 

1 
2 

1.4 
2.8 

94 

835 

850 

864 

879 

^894 

*9°9 

*923 

*938 

953 

967 

3 

A 

4.2 
6  a 

95 

982 

997 

*OI2 

*026 

*IOO 

*n4 

TE 

ft 

•  U 

7  O 

96 

47  129 

144 

159 

*73 

1  88 

202 

217 

232 

246 

261 

*J 

6 

1  •  Vf 

8.4 

97 

276 

290 

305 

319 

334 

349 

363 

378 

392 

407 

7 

9.8 

98 

422 

436 

451 

465 

480 

494 

509 

'524 

538 

553 

8 

11.2 

99 

567 

582 

596 

6n 

625 

640 

654 

669 

683 

698 

9 

12.6 

800 

712 

727 

741 

756 

770 

784 

799 

813 

828 

842 

N. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Pr 

»p.  Pts. 

TABLE  I.  " 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

0 

Prop.  Pts. 

800 

47  712 

727 

741 

756 

770 

784 

799 

813 

828 

842 

01 

857 

871 

885 

900 

914 

929 

943 

958 

972 

986 

02 

48  ooi 

015 

029 

044 

058 

073 

087 

101 

116 

130 

03 

144 

159 

173 

187 

202 

216 

230 

244 

259 

273 

15 

04 

287 

302 

316 

330 

344 

359 

373 

387 

401 

416 

1 

1.5 

05 
06 

430 
572 

AAA 
M  r*r 

586 

458 
601 

6if 

487 
629 

643 

515 
657 

530 
671 

in 

558 
700 

2 
3 

3.0 
4.5 

07 

714 

728 

742 

756 

770 

78J 

799 

813 

827 

841 

4 

6.0 

08 

855 

869 

883 

J*& 

911 

926 

940 

954 

968 

982 

5 

7.5 

09 

996 

*OIO 

*O24 

*o66 

*o8o 

*io8 

*I22 

6 

7" 

9.0 

1ft  K 

810 

49  136 

150 

164 

178 

192 

206 

220 

234 

248 

262 

8 

1U.O 

12.0 

11 

276 

290 

304 

318 

332 

346 

360 

374 

388 

402 

9 

13.5 

12 
13 

554 

429 
568 

443 
582 

457 
596 

471 
610 

485 
624 

513 
651 

679 

14 

693 

707 

721 

734 

748 

762 

776 

790 

803 

8l7 

'  15 

831 

845 

859 

872 

886 

900 

*9*4 

927 

941 

*9" 

14 

16 

969 

982 

996 

*OIO 

*024 

"037 

"065 

1 

1.4 

17 

50  106 

120 

133 

H7 

161 

174 

1  88 

202 

215 

229 

2 

2.8 

18 

243 

256 

270 

284 

297 

325 

338 

352 

365 

3 

4.2 

19 

379 

393 

406 

420 

433 

447 

461 

474 

488 

SOI 

4 

5.6 

7/v 

820 

515 

529 

542 

556 

569 

583 

596 

610 

623 

637 

6 

.0 
8  4 

21 
22 
23 

X 

920 

664 
799 
934 

678 
813 
947 

691 
826 
961 

70S 
840 

974 

718 
987 

732 
866 

*OOI 

745 
880 

759 
893 

*028 

772 
%907 

7 
8 
9 

9.8 
11.2 
12.6 

24 

51  055 

068 

08  1 

095 

1  08 

121 

135 

148 

162 

175 

25 
26 

188 
322 

202 

335 

348 

228 
362 

242 
375 

388 

402 

282 
415 

295 

308 
441 

27 

455 

468 

481 

495 

508 

521 

534 

548 

561 

574 

18 

28 

587 

601 

614 

627 

640 

654 

667 

680 

693 

706 

1 

1.3 

29 

720 

733 

746 

759 

772 

786 

799 

812 

825 

838 

2 

2.6 

880 

851 

865 

878 

891 

904 

917 

930 

943 

957 

970 

3 

3.9 

31 

983 

996 

"009 

*022 

*Q35 

*048 

*o6i 

*075 

*o88 

*IOI 

4 

5.2 

6K 

32 

52  114 

127 

140 

153 

1  66 

179 

192 

205 

218 

231 

•  O 

7Q 

33 

244 

257 

270 

284 

297 

310 

323 

336 

349 

362 

7 

.0 

9.1 

34 

375 

388 

401 

414 

427 

440 

453 

466 

479 

49* 

8 

10.4 

35 
36 

504 
634 

I17 
647 

543 
673 

556 
686 

569 
699 

582 
711 

595 
724 

608 
737 

621 
750 

9 

11.7 

37 

763 

776 

789 

802 

815 

827 

840 

853 

866 

*879 

38 

892 

905 

917 

930 

943 

956 

969 

982 

994 

39 

53  020 

033 

046 

058 

071 

084 

097 

no 

122 

135 

12 

840 

148 

161 

173 

1  86 

199 

212 

224 

237 

250 

263 

1 

1.2 

41 

275 

288 

301 

3H 

326 

339 

352 

364 

377 

390 

2 

2.4 

3/> 

42 

403 

415 

428 

441 

466 

479 

491 

504 

.6 

43 

529 

542 

555 

567 

580 

593 

605 

618 

631 

643 

4 
5 

4.8 
6  0 

44 
45 

656 
782 

668 
794 

681 
807 

694 
820 

706 
832 

719 
845 

732 
857 

744 
870 

882 

769 
895 

6 

7 

7.2 
8.4 

46 

908 

920 

933 

945 

958 

970 

983 

995 

*oo8 

*020 

8 

9.6 

47 

54033 

045 

058 

070 

083 

095 

108 

120 

133 

145 

9 

10.8 

48 

158 

170 

183 

195 

208 

220 

233 

245 

258 

270 

49 

283 

295 

307 

320 

332 

345 

357 

370 

382 

394 

850 

407 

419 

432 

444 

456 

469 

481 

4941 

506 

518 

N. 

0 

1 

2 

3 

4 

5 

6 

7  1  8 

9 

Prop.  Pts. 

LOGARITHMS  OF  NUMBERS. 


N. 

BMMMMM 

350 

51 
52 
53 

54 
,  55 
56 

57 
58 
59 

360 

61 
62 
63 

64 
65 
66 

67 
68 
69 

370 

71 
72 
73 

74 
75 
76 

77 
78 
79 

880 

81 
82 
83 

84 
85 
86 

87 

88 
89 

390 

91 

92 
93 

94 
95 
96 

97 
98 
99 

400 

N. 

O 

MHMMMMMMM 

54  407 

lj_ 

419 

2 

432 

3 

444 

4 

456 

5 

469 

6 

481 

7 

494 

8 
~ 

9 

~ 

Prc 

<MMMMKH 

1 

2 
3 
4 
5 
6 
7 
8 
9 

1 
2 
3 
4 
5 
6 
7 
8 
9 

1 
2 
3 
4 
1 
6 
7 
S 
1 

1 
2 
3 
4 
5 
6 
7 
8 
9 

~Pro 

>p.  Pts. 

»—  —  —  — 

13 

1.3 
2.6 
3.9 
5.2 
6.5 
7.8 
9.1 
10.4 
11.7 

12 

1.2 
2.4 

3.6 
4.8 
6.0 
7.2 
8.4 
9.6 
10.8 

11 

1.1 
2.2 
3.3 
4.4 
5.5 
6.6 
7.7 
8.8 
9.9 

10 

1.0 
2.0 
3.0 
4.0 
5.0 
6.0 
7.0 
8.0 
9.0 

p.  Pts. 

53i 
654 

777 
900 
55  023 
145 
267 
388 
509 
630 

75i 
871 
991 

56  1  10 

467 
585 
703 

543 
667 
790 

913 
035 
157 

279 
400 
522 

555 
679 
802 

925 
047 
169 

291 
4i3 

534 

568 
691 
814 

937 
060 
182 

303 

425 
546 

580 
704 
827 

949 
072 
194 

315 

437 
558 

593 
716 

839 
962 
084 
206 

328 

449 
570 

605 
728 
851 

974 
096 
218 

340 
461 
582 

617 
741 
864 

986 
1  08 
230 

352 

473 
594 

630 

753 
876 

998 

121 

242 

364 
485 
606 

642 

765 
888 

*OII 

133 
255 

376 
497 
618 

642 

654 

666 

678 

691 

703 

715 

727 

739 

763 
883 
*OO3 

122 
241 
360 

478 

597 
7H 

775 
895 
*oi5 

134 
253 
372 

490 
608 
726 

787 
907 

*027 

146 

265 
384 
502 

620 

738 

799 
919 
"038 

158 

277 
396 

5H 
632 
750 

811 

93i 
*o5o 

170 
289 

407 

526 

644 
761 

823 
*9J3 

*062 

182 
301 
419 

1% 

656 

773 

835 
955 
"074 

194 
312 
43i 

549 
667 

785 

847 
967 

*o86 

205 

324 
443 
561 
679 
797 

859 
979 
*098 

217 
336 
453 

573 
691 
808 

820 

832 

844 

855 

867 

879 

891 

902 

914 

926 

937 
57  054 
171 

287 
403 
519 

634 

749 
864 

978 

9i? 
066 

183 

299 
415 
530 

646 
76i 
875 

961 
078 
194 

310 
426 
542 

657 
772 
887 

972 
089 
206 

322 
438 
553 

669 

784 
898 

984 

101 

217 

334 
449 
565 
680 

795 
910 

996 

"3 
229 

345 
461 

576 

692 
807 
921 

*oo8 
124 
241 

357 
473 
588 

8°8 
933 

*oi9 
136 
252 

368 

484 
600 

715 
830 

944 

"031 
148 
264 

380 
496 
61-1 

726 
841 
955 

*043 

159 
276 

392 
507 
623 

738 
8f 
967 

990 

*OOI 

*OI3 

*O24 

*Q35 

*047 

"058 

*O7O 

*o8i 

58  092 
206 
320 

433 
546 
659 

771 
883 
995 

104 
218 
331 

444 
557 
670 

782 

894 
*oo6 

"5 

229 

343 

456 
569 

681 

794 
906 

*OI7 

127 
240 

354 
467 

e 

805 
917 

*028 

138 
252 

365, 
478 

591 
704 

816 
928 
*O4o 

149 
263 

377 

490 
602 
715 
827 

939 
*o5i 

161 
274 
388 

501 
614 
726 

838 
« 

172 
286 
399 
512 
625 
737 
850 
961 
*073 

184 
297 
410 

524 
636 

749 
861 

973 
*o84 

195 
309 
422 

I3* 
647 

760 

872 
984 
*o9s 

59  106 

118 

129 

140 

151 

162 

173 

184 

195 

207 

218 
329 
439 

770 
879 

60  097 

229 

340 
450 

I61 
671 

780 
890 

240 
461 

III 

791 
901 

*OIO 

119 

2I' 
362 

472 

a 

802 
912 

*02I 
130 

262 
373 
483 

594 
704 

813 

923 
"032 
141 

273 
384 
494 
605 

715 
824 

934 
"043 
152 

284 

395 
506 

616 
726 
835 

945 
*Q54 
163 

295 
406 

517 
627 
737 
846 

956 
*o6ij 
173 

306 

417 
528 

638 
748 
857 

966 
"076 
184 

3i8 
428 

539 
649 

759 
868 

977 
*o86 

195 

206 
O 

217 

mfmm^mmmm 

1 

228 
2 

239 

3 

249 
4 

260 
5 

271 
6 

282 
7 

293 

8 

304 
9 

TABLE  I. 


1  N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

400 

60  206 

217 

228 

239 

249 

260 

271 

282 

293 

304 

01 

3H 

325 

336 

347 

358 

369 

379 

390 

401 

412 

02 

423 

433 

444 

455 

466 

477 

487 

498 

509 

520 

03 

531 

541 

552 

563 

574 

584 

595 

606 

617 

627 

04 

638 

649 

660 

670 

68  1 

692 

703 

713 

724 

735 

05 

746 

756 

767 

778 

788 

799 

810 

821 

83'  842 

06 

853 

863 

874 

885 

895 

906 

917 

927 

938 

949 

11 

07 

959 

970 

981 

991 

*O02 

"013 

*023 

*034 

*045 

*055 

1 

1.1 

08 

61  066 

077 

087 

098 

I09 

119 

130 

140 

151 

162 

2 

2.2 

09 

172 

183 

194 

204 

215 

225 

236 

247 

257 

268 

3 

A 

3.3 

A  A 

410 

278 

289 

300 

310 

321 

33i 

342 

352 

363 

374 

•I 

5 

tb.4 

5.5 

11 

384 

395 

405 

416 

426 

437 

448 

458 

469 

479 

6 

6.6 

12 
13 

490 
595 

500 
606 

5" 
616 

SJ 

532 
637 

542 
648 

553 
658 

g 

I74 
679 

584 
690 

7 
8 

7.7 
8.8 

14 
15 

700 
805 

711 
815 

721 
826 

731 
836 

742 
847 

752 
857 

763 
868 

III 

784 
888 

794 
899 

9 

9.9 

16 

909 

920 

930 

941 

951 

962 

972 

982 

993 

*c»3 

17 

62  014 

024 

034 

045 

055 

066 

076 

086 

097 

107 

18 

118 

128 

138 

149 

159 

170 

1  80 

190 

201 

211 

19 

221 

232 

242 

252 

263 

273 

284 

294 

304 

315 

420 

325 

335 

346 

356 

366 

377 

387 

397 

408 

4l8 

21 

428 

439 

449 

459 

469 

480 

490 

500 

511 

521 

IV 

22 
23 

531 
634 

542 
644 

IS 

562 
665 

572 
675 

583 
685 

603 
706 

716 

624 
726 

1 
2 

1.0 
2.0 

24 

737 

747 

757 

767 

778 

788 

798 

808 

818 

829 

3 

3.0 

4   A 

25 
26 

839 
941 

849 
95i 

s? 

870 
972 

880 
982 

890 
992 

900 

*002 

910 

*OI2 

921 

*022 

931 
*033 

4 
5 
6 

.0 
5.0 
6.0 

27 

63  043 

053 

063 

073 

083 

094 

IO4 

114 

124 

134 

7 

7.0 

28 

144 

155 

165 

175 

185 

195 

205 

215 

225 

236 

8 

8.0 

29 

246 

256 

266 

276 

286 

296 

306 

317 

327 

337 

9 

9.0 

430 

347 

357 

367 

377 

387 

397 

407 

417 

428 

438 

31 

448 

458 

468 

478 

488 

498 

508 

5l8 

528 

538 

32 

548 

558 

568 

579 

589 

599 

609 

619 

629 

639 

33 

649 

659 

669 

679 

689 

699 

709 

719 

729 

739 

34 

35 

849 

759 
859 

769 

869 

779 
879 

789 
889 

899 

809 
909 

819 
919 

829 
929 

839 
939 

36 

949 

959 

969 

979 

988 

998 

*oo8 

*oi8 

*028 

*o38 

9 

37 

64  048 

058 

068 

078 

088 

098 

108 

118 

128 

137 

1 

0.9 

38 

147 

157 

167 

177 

187 

197 

207 

217 

227 

237 

2 

1.8 

39 

246 

256 

266 

276 

286 

296 

306 

316 

326 

335 

3 

2.7 

440 

345 

355 

365 

375 

385 

395 

404 

414 

424 

434 

4 

3.6 

4C 

41 

444 

454 

464 

473 

483 

493 

503 

513 

523 

532 

6 

.0 

5.4 

42 

542 

552 

562 

572 

582 

59i 

601 

611 

621 

631 

7 

ft  a 

43 

640 

650 

660 

670 

680 

689 

699 

709 

719 

729 

1 

8 

u  •  %} 

7.2 

44 

738 

748 

758 

768 

777 

787 

797 

807 

816 

826 

9 

8.1 

45 

836 

846 

856 

865 

875 

885 

904 

914 

924 

46 

933 

943 

953 

963 

972 

982 

992 

*O02 

*OII 

*O2I 

47 

65  031 

040 

050 

060 

070 

079 

089 

099 

108 

118 

48 

128 

137 

H7 

157 

167 

176 

1  86 

I96 

205 

215 

^  \ 

49 

225 

234 

244 

254 

263 

273 

283 

292 

302 

312 

450 

321 

33i 

341 

350 

360 

369 

379 

389 

398 

408 

N. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

LOGARITHMS  OF  NUMBERS. 


N. 

450 

51 
52 
53 

54 
55 
56 

57 
58 
59 

460 

61 
62 
63 

64 
65 
66 

67 

68 
69 

470 

71 
72 
73 

74 
75 
76 

77 
78 
79 

480 

81 
82 
83 

84 
85 
86 

87 
88 
89 

490 

91 
92 
93 

94 
95 
90 

97 
93 
99 

500 

O 

65  321 

1 

2 

•••MM^M 

341 

3 

350 

4 

5 

~ 

6 

•••••»-» 

379 

7 

m*mmm*mmm 

389 

8 
"398" 

9 

408 

Pro] 

••••••••••i 

1 
2 
3 
4 
5 
6 
7 
8 
9 

1 
2 
3 
4 
5 
6 
7 
8 
9 

1 

2 
3 
4 
5 
6 
7 
8 
9 

•IMMHHMB 

Pro 

).  PtS. 

•MMMi^M 

10 

1.0 
2.0 
3.0 
4.0 
5.0 
6.0 
7.0 
8.0 
9.0 

9 

0.9 
1.8 
2.7 
3.6 
4.5 
5.4 
6.3 
7.2 
8.1 

8 

0.8 
1.6 
2.4 
3.2 
4.0 
4.8 
5.6 
6.4 
7.2 

p.  Pts. 

418 

5H 
610 

706 
801 
896 

992 
66  087 
181 

276 

370 
464 
558 
652 
745 
839 

932 
67  025 
117 

427 

523 
619 

715 
811 
906 

*OOI 

096 
191 

437 
533 
629 

725 
820 
916 

*OII 

106 
200 

447 
543 
639 

734 
830 

925 

*O2O 
2IO 

456 

III 

744 
839 
935 

124 
219 

466 
562 
658 

753 
849 
944 

134 
229 

475 
57i 
667 

763 
858 

954 
238 

485 

|8l 
677 

772 

868 
963 

153 
247 

495 
686 

782 
877 
973 
*o68 
162 
257 

504 
600 
696 

792 
887 
982 

172 

266 

285 

295 

304 

3T4 

323 

332 

342 

351 

361 

380 
474 
567 
661 

755 
848 

941 

034 
127 

389 
483 
577 

671 
764 
857 
950 

043 
136 

398 
492 
586 

680 

773 
867 

960 
052 
MS 

408 
502 
596 

689 

783 
876 

969 
062 
154 

417 

33 

699 
792 
885 

978 
071 
164 

427 
521 
614 

708 
801 
894 

987 
080 
173 

436 
530 
624 

717 
811 
904 

997 
089 
182 

445 
539 
633 

727 
820 

913 
*oo6 
099 
191 

455 
549 
642 

736 
829 
922 

20  1 

210 

219 

228 

237 

247 

256 

265 

274 

284 

293 

302 

394 
486 

H8 
669 

761 
852 

*o  943 

68  034 

3" 
403 
495 

587 
679 
770 

861 
952 
043 

321 

504 

596 
688 
779 
870 
961 
052 

330 
422 

605 

788 

879 
970 
06  1 

339 

523 
614 
706 
797 
888 

979 
070 

348 
440 
532 
624 

897 
988 
079 

357 
449 
541 

633 
724 

906 

997 
088 

367 
459 
550 
642 

733 
825 

916 
*oo6 
097 

376 
468 
560 

651 
742 
834 

925 
*oi5 
106 

385 
477 
569 

660 

752 
843 

934 
"024 

124 

133 

142 

I5I 

160 

169 

178 

187 

196 

205 

215 
305 
395 

485 
574 
664 

753 
842 

224 

3H 
404 

494 
583 
673 

762 
851 
940 

233 
323 
413 
502 
592 
68  1 

771 
860 
949 

242 
332 
422 

601 
690 

780 
869 
958 

251 

431 
520 
610 
699 

789 
878 
966 

260 
350 
440 

529 
619 
708 

975 

269 
359 
449 

538 
628 
717 

806 
895 
984 

278 
368 
458 

547 
637 
726 

815 

904 

993 

287 
377 
467 

SA 
646 

735 
824 
913 

*OO2 

$ 

st 

744 

833 
922 

*OII 

69  020 

028 

037 

046 

055 

064 

073 

082 

090 

099 

108 
197 
285 

3P 
461 

548 
636 

810 
897 

^^MHHHMHBBBMC 

O 

117 

205 
294 

469 
557 
644 
732 
819 

126 
214 
302 

390 
478 
566 

653 
740 
827 

135 
223 

399 
487 
574 
662 

749 
836 

144 
232 
320 

408 
496 
583 
671 
758 
845 

152 
241 
329 

417 
504 

592 
679 
767 
854 

161 
249 
338 
425 

513 
601 

688 

& 

170 
258 
346 

434 
522 
609 

697 
784 
871 

179 
267 

355 

443 

705 

793 
880 

1  88 
276 
364 

452 
539 
627 

7H 
801 
888 

906 

914 

923 
3 

932 
4 

940 
5 

949 
6 

958 
7 

966 

975 

N. 

1 

2 

8 

9 

10 


TABLE  I. 


N. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

500 

01 
02 
03 

04 
05 
06 

07 

08 
09 

510 

11 
12 
13 

14 
15 
16 

17 
18 
19 

520 

21 
22 
23 

24 
25 
26 

27 

28 
29 

580 

31 
32 
33 

34 
35 
36 

37 
38 
39 

540 

41 
42 
43 

44 
45 
46 

47 

48 
49 

550 
N. 

69897 

984 
70  070 

157 

243 
329 
415 
501 
586 
672 

906 

914 

923 

932 

940 

949 

958 

966 

975 

1 
2 
3 
4 
5 
6 
7 
8 
9 

1 
2 
3 
4 
5 
6 
7 
8 
9 

1 
2 
3 
4 
5 
6 
7 
8 
9 

^M^BM^ 

Pro 

9 

0.9 
1.8 
2.7 
3.6 
4.5 
5.4 
6.3 
7.2 
8.1 

8 

0.8 
1.6 
2.4 
3.2 
4.0 
4.8 
5.6 
6.4 
7.2 

7 

0.7 
1.4 
2.1 
2.8 
3.5 
4.2 
4.9 
5.6 
6  3 

P.  Pts. 

992 
079 
165 

252 
338 
424 

509 

595 
680 

*OOI 

088 
174 
260 
346 
432 

li8 
603 

689 

*OIO 

096 
183 
269 

355 

441 

526 
612 
697 

*oi8 
105 
191 

278 
364 
449 

I35 
621 

706 

*027 
114 
2OO 

286 
372 
458 

544 
629 

7H 

*036 

122 
.209 

295 

381 
467 

III 

723 

*044 
131 
217 

303 
389 
475 
561 
646 
73i 

*Q53 
140 
226 

312 
398 
484 
569 

655 
740 

*062 

148 
234 

321 
406 

492 

578 
663 

749 

757 
842 
927 

71  012 
096 

181 
265 

(  349 
433 
517 

766 

851 
935 

020 

105 
189 
273 

357 
441 

525 

774 

783  ;  791 

800 

808 

817 

825 

834 

859 
944 
029 

113 

366 
450 
533 

868 
952 
037 

122 
206 
290 

374 
458 
542 

876 
961 
046 

130 
214 

299 

383 
466 

550 

885 
969 
054 

139 

223 

307 

391 
475 
559 

893 
978 
063 

147 
231 

315 

399 
483 
567 

902 
986 
071 

155 

240 

324 
408 
492 
575 

910 

995 
079 

164 
248 
332 

416 
500 
584 

919 
*oo3 
088 

172 
257 
341 

s 

592 

600 

609 

617 

625 

634 

642 

650 

659 

667 

675 

684 
767 
850 

933 
72  016 
099 

181 
"^28" 

692 

III 

941 
024 
107 

189 
272 
354 

700 
784 
867 

950 
032 
"5 

362 

709 
792 
875 

958 
041 
123 

206 
288 
370 

717 
800 
883 

966 
049 
132 

214 
296 
378 

725 
809 
892 

975 
057 
140 

222 
304 
387 

734 
817 
900 

983 
148 

230 
313 
395 

742 

8 

991 
074 
156 

239 
321 
403 

750 

834 
917 

163 

247 
329 
411 

759 
842 
925 

*oo8 
090 
173 

255 
337 
419 

436 

444 

452 

460 

469 

477 

485 

493 

501 

509 
59i 
673 

754 
835 
916 

997 
73  078 
159 

518 

599 
681 

762 

843 
925 

*oo6 
086 
167 

526 
607 
689 

770 
852 
933 
*oi4 
094 
175 

I3! 

616 

697 

779 
860 

941 

*022 
102 
I83 

542 
624 
705 

787 
868 

949 
*03o 
in 
191 

1S° 
632 

713 

795 
876 

957 

*038 
119 
199 

558 
640 
722 

803 
884 
965 

*046 
127 
207 

567 
648 
730 

811 

892 
973 
*054 
135 
215 

575 
656 

738 

819 
900 
981 

*062 

143 
223 

583 
665 
746 

827 
908 
989 

*O7O 

151 
231 

239 

247 

255 

263 

272 

280 

288 

296 

304 

312 

320 
400 
480 

560 
640 
719 

1% 

957 

74  036 

———•-• 

O 

328 
408 
488 

568 
648 
727 

807 
886 
965 

336 
416 
496 

i^ 

656 

735 
815 
894 
973 

344 
424 
504 

584 
664 

743 

823 
902 
981 

352 
432 
512 

592 
672 

751 
830 
910 
989 

360 
440 
520 

600 
679 
759 

838 
918 
997 

368 
448 
528 

608 
687 
767 

846 
926 
*oo3 

376 
456 
536 

616 
695 
775 

854 
933 
*oi3 

384 

464 

544 
624 
703 
783 
862 

*941 

*O2O 

392 
472 
552 

632 
711 
791 

870 
949 

*028 

044 

«MMM»M» 

1 

052 
2 

060 
3 

068 
—  — 
4 

076 

^••MiMB 

5 

084 
6 

092 
7 

099 

8 

107 

9 

LOGARITHMS  OF  NUMBERS. 


1 1 


fmmmmv^ 

N. 

550 

5.1 
52 
53 

54 
55 
56 

57 
58 
59 

660 

61 

62 
63 

64 
65 
66 

67 
68 
69 

570 

71 
72 
73 

74 
75 
76 

77 
78 
79 

580 

81 
82 
83 

84 
85 
86 

87 

88 
89 

590 

91 
92 
93 

94 
95 
96 

97 

98 
99 

600 

N. 

mmm^mttmrmm 

0 

74  036 

§•••••• 

1 
044 

2 

052 

3 

060 

4 

•^M^^M 

068 

5 

^^MHMK 

O76 

6 

084 

r 

092 

8 
099 

9 

107 

Pro] 

••^•M^H 

1 

2 
3 
4 
5 
6 
7 
8 
9 

1 
2 
3 
4 
5 
6 
7 
8 
9 

"Pr7 

>.Pte. 

««••••• 

8 

0.8 
1.6 
2.4 
3.2 
4.0 
4.8 
5.6 
6.4 
7.2 

7 

0.7 
1.4 
2.1 
2.8 
3.5 
4.2 
4.9 
5.6 
6.3 

p.Pts. 

US 
194 

273 

351 
429 
507 

586 
663 

74i 

123 

202 
280 

359 
437 
515 

593 
671 
749 

131 

210 
288 

367 

445 
523 
601 
679 

757 

139 
218 
296 

374 
453 
531 
609 
687 
764 

147 
225 

304 
382 
46l 

539 
617 
695 
772 

155 
233 
312 

390 
468 

547 

624 
702 
780 

162 
241 
320 

398 
476 

554 

632 
710 
788 

170 
249 
327 

406 

484 
562 

640 
718 
796 

178 
257 
335 
414 
492 
570 

648 
726 
803 

1  86 
265 
343 

421 
500 
578 
656 

733 

SH 

819 

827 

834 

842 

850 

858 

865 

873 

881 

889 

896 

974 
75051 

128 
205 
282 

358 
435 
5" 

904 
981 
059 

136 
213 
289 

366 
442 
519 

912 

989 
066 

143 

220 

297 

374 
450 
526 

920 

997 
074 

I5i 

228 

305 

38i 
458 
534 

927 
*ool 
082 

'59 
236 
312 

389 
465 

542 

935 

*OI2 
089 

166 

243 
320 

397 
473 
549 

943 

*020 

097 

174 

251 

328 

404 
481 
557 

950 

*028 

105 

182 

259 

335 
412 
488 
565 

958 
*035 
"3 
189 
266 
343 
420 
496 
572 

966 
*043 

120 

197 
274 
351 

427 
504 
580 

587 

595 

603 

610 

618 

626 

633 

641 

648 

656 

664 
740 
815 

891 
967 
76  042 

118 

193 
268 

343 

671 

747 
823 

899 

974 
050 

125 
200 
275 

679 
755 
83' 
906 
982 
057 

133 
208 

283 

686 
762 
-838 

914 

989 
o6J 

140 
215 

290 

694 
770 
846 

921 

997 
072 

148 
223 
298 

702 
778 
853 

929 
*oos 
080 

155 
230 

305 

709 

785 
861 

937 

*OI2 
087 

163 
238 
313 

717 

793 
868 

944 

*020 
095 

170 

245 
320 

724 
800 
876 

952 

*O27 

103 
178 

253 
328 

732 
808 

884 

959 
*<>35 
no 

'£ 

260 
335 

350 

358 

365 

373 

380 

388 

395 

403 

410 
486" 

I59 
634 

708 
782 
856 

930 
*oo4 
078 

418 
492 
567 

641 
716 
790 

77  012 

425 
500 

574 
649 
723 
797 

871 

945 
019 

433 
507 
582 

656 

730 
805 

879 
953 
026 

440 
515 
589 
664 
738 
812 

886 
960 
034 

448 
522 

597 
671 

745 
819 

^ 
967 

041 

455 
530 
604 

678 

753 
827 

901 
975 

462 

537 
612 

686 
760 
834 
908 

982 

056 

470 

545 
619 

693 
768 
842 

916 

989 
063 

477 

g 

% 

849 

923 
997 
070 

085 

093 

IOO 

107 

H5 

122 

129 

137 

144 

151 

&  159- 
232 
305 

379 
452 
525 

597 
670 

743 

166 
240 
313 
386 
459 
532 

605 
677 
750 

173 
247 
320 

3?I 
466 

539 
612 
685 
757 

181 
254 
327 
401 

474 
546 

619 
692 
764 

188 
262 
335 
408 
481 
554 

627 
699 
772 

195 
269 

342 

415 
488 
56l 

634 
706 

779 

203 
276 
349 
422 

495 
568 

641 

7H 
786 

210 
283 

357 

430 
5«>3 
576 

648 
721 
793 

217 
291 
364 

437 
5io 

583 
656 
728 
801 

225 

371 

444 
517 
590 

663 
78ol 

815 

^••I^HH^^B^B 

0 

822 

1 

830 
3 

837 
3 

844 
4 

851 
5 

859 
6 

866 
7 

873 

8 

880 
9 

12 


TABLE  I. 


N. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Pro] 

E>.Pts. 

600 

77  815 

822 

830 

837 

844 

851 

859 

866 

873 

880 

01 
02 
03 

04 
05 
06 

07 
08 
09 

887 
960 
78  032 

104 
176 
247 

319 
390 
462 

895 
967 
039 
in 
183 
254 

326 
398 
469 

902 

974 
046 

118 
190 

262 

333 
405 
476 

909 
981 
053 
125 
197 
269 

340 
412 

483 

916 
988 
061 

132 

204 
276 

347 
419 
490 

924 
996 
068 

140 

211 
283 

355 
426 

497 

*931 
*oo3 

075 

147 
219 
290 

362 

433 
504 

*938 

*OIO 

082 

154 
226 
297 

369 
440 
512 

945 
*oi7 
089 

161 
233 
305 

376 

447 
519 

952. 

*025 

097 

1  68 
240 
312 

383 

455 
526 

1 
2 

3 

8 

0.8 
1.6 
2.4 

39 

610 

533 

540 

547 

554 

561 

569 

576 

583 

590 

597 

5 

4.0 

11 
12 
13 

14 
15 
16 

17 

18 
19 

604 
675 
746 

817 
888 
958 

79  029 

099 
169 

611 
682 
753 

824 

895 
965 

036 
1  06 
176 

618 
689 
760 

831 
902 

972 

043 
H3 
183 

625 
696 
767 

838 
909 

979 
050 
1  20 
190 

633 
704 

774 

845 
916 

986 

057 

127 
197 

640 
711 
78i 
852 
923 
993 
064 

134 
204 

647 
718 
789 

859 
930 

*000 

071 
141 

211 

654 
725 
796 

866 

937 
*oo7 

078 
148 
218 

661 
732 
803 

873 

*944 
*oi4 

085 

155 

225 

668 

739 
810 

880 
95i 

*02I 

092 
162 
232 

6 
7 
8 
9 

4.8 
5.6 
6.4 
7.2 

620 

239 

246 

253 

260 

267 

274 

28l 

288 

295 

302 

21 
22 
23 

24 
25 
26 

27 
28 
29 

309 
379 
449 
518 
588 
657 

727 
796 
865 

316 
386 
456 

525 
595 
664 

734 
803 
872 

323 
393 
463 

532 
602 
671 

741 
810 

879 

330 
400 
470 

539 
678 

748 
817 
886 

337 
407 

477 

546 
616 
685 

754 
824 

893 

344 
414 
484 

553 
623 
692 

761 
831 
900 

351 

421 
491 

S60^ 
630 
699 

768 

837 
906 

358 
428 

498 
567 

637 
706 

775 
844 

913 

365 
435 
505 

I74 
644 

713 
782 
851 
920 

372 
442 
5ii 

58i 
650 
720 

789 
858 
927 

1 
2 
3 
4 
5 
C 
7 
8 
9 

7 

0.7 
1.4 
2.1 
2.8 
3.5 
4.2 
4.9 
5.6 
6.3 

630 

934 

941 

948 

955 

962 

969 

975 

982 

989 

996 

- 

31 
32 
33 

34 
1  35 
36 

37 
38 
39 

80  003 
072 
140 

209 

277 
346 

414 
482 
550 

OIO 

079 
H7 

216 

284 
353 
421 
489 
557 

017 
085 
154 

223 
291 

359 
428 
496 

564 

024 
092 
161 

229 
298 
366 

434 
502 
570 

030 
099 
1  68 

236 
305 

373 
441 
509 
577 

037 
106 

175 

243 

312 
380 

448 
516 

584 

044 

H3 
182 

250 
3i8 
387 

455 
523 
59i 

051 

120 

188 

257 
325 

393 
462 
530 
598 

058 
127 
195 
264 
332 
400 

468 
536 
604 

065 
134 

202 
271 

339 
407 

475 
543 
611 

1 
2 
3 

6 

0.6 

1.2 
1.8 

640 

618 

625 

632 

638 

645 

652 

659 

665 

672 

679 

4 

2.4 

41 
42 
43 

44 
45 
46 

47 

48 
49 

686 

754 
821 

889. 
956 
8  1  023 

090 
158 

224 

693 
760 
828 

895 
963 
030 

097 
164 
231 

699 
767 
835 
902 
969 
037 
104 
171 
238 

706 

774 
841 

909 
976 
043 
in 
178 
245 

713 

781 
848 

916 

983 
050 

117 
184 
251 

720 
787 
855 
922 
990 
057 

124 
191 
258 

726 

794 

862 

929 

996 

064 

I3J 
198 
265 

733 
801 
868 

93<3 
*oo"j 
070 

137 

204 
271 

740 
808 
875 

*943 

*OIO 

077 
144 

211 

278 

747 
814 
882 

949 
*oi7 

084 

151 

218 

285 

5 
6 
7 
8 
9 

3.0 
3.6 
4.2 
4.8 
5.4 

650 
N. 

291 
O 

298 

1 

305 
2 

3ii 
3 

318 
4 

325 
5 

33i 
6 

338 

KMWMM 

7 

345 
8 

351 
9 

Pro 

p  Pts. 

LOGARITHMS  OF  NUMBERS. 


N. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

650 

81  291 

298 

305 

311 

3i8 

325 

33i 

338 

345 

35i 

51 

358 

365 

371 

378 

385 

391 

398 

405 

411 

418 

52 

425 

438 

445 

458 

465 

47i 

478 

485 

53 

491 

498 

505 

5i8 

525 

538 

544 

551 

54 
55 

558 
624 

564 
631 

637 

578 
644 

584 
651 

657 

Li 

604 
671 

611 
677 

617 
684 

56 

690 

697 

704 

710 

717 

723 

730 

737 

743 

750 

57 

757 

763 

770 

776 

783 

790 

796 

803  809 

816 

58 

823 

829 

836 

842 

849 

856 

862 

869 

875 

882 

59 

889 

895 

902 

908 

915 

921 

928 

935 

941 

948 

660 

954 

961 

968 

974 

981 

987 

994 

*ooo 

*oo7 

*oi4 

61 

82  020 

027 

033 

040 

046 

053 

060 

066 

073 

079 

62 

086 

092 

099 

105 

112 

"9 

125 

132 

138 

143 

1 

0.7 

63 

151 

158 

164 

171 

I78 

184 

191 

197 

204 

210 

2 

1.4 

64 
65 

217 
282 

223 
289 

230 
295 

236 
302 

308 

249 
315 

256 
321 

263 
328 

269 
334 

276 
341 

3 

4 

2.1 
2.8 

3e 

66 

347 

354 

36o 

367 

373 

380 

387 

393 

400 

406 

6 

•  O 

4.2 

67 

413 

419 

426 

432 

439 

445 

452 

458 

465 

471 

7 

4.9 

68 

478 

484 

491 

497 

504 

510 

523 

530 

536 

8 

5.6 

69 

543 

549 

556 

562 

569 

575 

582 

588 

595 

601 

9 

6.3 

670 

607 

614 

620 

627 

633 

640 

646 

653 

659 

666 

71 

672 

679 

685 

692 

698 

705 

711 

718 

724 

730 

72 

73 

737 
802 

808 

750 
814 

756 
821 

763 
827 

769 
834 

776 
840 

782 
847 

789 
853 

795 
860 

74 

866 

872 

879 

885 

892 

898 

905 

911 

918 

924 

75 

930 

937 

943 

*95° 

956 

963 

*969 

975 

982 

988 

76 

995 

*OOI 

*oo8 

*O2O 

*027 

*046 

"052 

77 

83  059 

065 

072 

078 

085 

091 

097 

104 

no 

117 

78 

W  123 

129 

136 

142 

149 

155 

161 

168 

174 

181 

79 

187 

193 

200 

206 

213 

219 

225 

232 

238 

241 

680 

251 

257 

264 

270 

276 

283 

289 

296 

302 

308 

81 

315 

321 

327 

334 

340 

347 

353 

359 

366 

372 

6 

82 
83 

378 
442 

448 

391 
455 

398 
461 

404 
467 

410 
474 

480 

423 
487 

429 
493 

436 
499 

1 
2 

0.6 
1.2 

84 

506 

512 

518 

525 

531 

537 

544 

55° 

556 

563 

3 

1.8 

2   A 

85 
86 

569 
632 

575 
639 

645 

588 
651 

594 
658 

664 

607 
670 

613 
677 

620 
683 

626. 
689 

5 
6 

.4 
3.0 
3.6 

87 

696 

702 

708 

715 

721 

727 

734 

740 

746 

753 

7 

4.2 

88 

759 

765 

771 

778 

784 

790 

797 

809 

816 

8 

4.8 

89 

822 

828 

835 

841 

847 

853 

860 

866 

872 

879 

0 

5.4 

690 

885 

891 

897 

904 

910 

916 

923 

929 

935 

942 

91 

948 

954 

960 

967 

973 

979 

985 

992 

998 

*oo4 

92 

84  on 

017 

023 

029 

036 

042 

048 

055 

061 

067 

93 

073 

080 

086 

092 

098 

105 

in 

117 

123 

130 

94 
95 

136 

142 
205 

148 

211 

155 
217 

161 
223 

167 
230 

III 

180 
242 

186 
248 

192 

251 

96 

261 

267 

273 

280 

286 

292 

298 

305 

3" 

317 

97 

98 

$ 

330 
392 

336 
398 

342 

404 

348 
410 

354 
417 

36i 
423 

367 
429 

373 

435 

379 

442 

99 

448 

454 

460 

466 

473 

479 

485 

491 

497 

504 

700 

510 

516 

522 

528 

535 

54i 

547 

553 

559 

566 

N. 

O 

1 

2 

3 

4 

•5 

6 

7 

8 

9 

Prop.  Pts. 

TABLE  I. 


N. 

0 

i 

2 

3 

4 

5 

6 

7 

8 

0 

Prop.  Pts. 

700 

01 
1   02 
03 

1  04 
05 
OG 

i  07 
1  08 
09 

710 

11 
12 
13 

14 
15 
16 

17 
18 
19 

720 

21 
22 
23 

24 

25 
26 

27 
28 
29 

780 

31 
32 
33 

34 

35 
36 

37 

38 
39 

740 

41 
42 
43 

44 
45 
46 

47 
48 
49 

750 
N. 

84  510 

516 

522 

528 

535 

541 

547 

553 

559 

566 

1 
2 
3 
4 
5 
6 
7 
8 
9 

1 
2 
3 
4 
5 
6 
7 
8 
9 

1 
2 
3 
4 
5 
6 
7 
8 
9 

••MI^M 

Pro] 

7 

0.7 
1.4 
2.1 
2.8 
3.5 
4.2 
4.9 
5.6 
6.3 

• 

0.6 
1.2 
1.8 
2.4 
3.0 
3.6 
4.2 
4.8 
5.4 

5 

0.5 
1.0 
1.5 
2.0 
2.5 
3.0 
3.5 
4.0 
4.5 

).PtS. 

572 

634 
696 

757 
819 
880 

942 
85  003 
065 
126 

I78 
640 

702 

763 
825 
887 

948 
009 
071 

584 
646 
708 

770 
831 
893 

954 
016 
077 

590 
652 
7H 

776 
837 
899 
960 

022 
083 

597 
658 
720 

782 
844 
905 

967 
028 
089 

603 
665 
726 

788 
850 
911 

973 
034 
095 

609 
671 
733 

794 
856 
917 

979 
040 

101 

6i5 

677 

739 
800 
862 
924 

107 

621 
683 
743 

807 
868 
93<> 
991 
052 
114 

628 
689 
75i 

8i3 
874 
936 

997 
058 
1  20 

132 

138 

144 

150 

156 

163 

169 

175 

181 

187 
248 
309 
370 

43i 
491 

552 
612 

673 

193 

254 

315 

376 

437 
497 

558 
618 
679 

199 
260 
321 

382 
443 
503 

564 
625 
685 

205 
266 
327 
388 

449 
509 

570 

631 
691 

211 

272 

333 

394 
455 
516 

576 
637 
697 

217 
278 
339 
400 
461 
522 

582 
643 
703 

224 
285 
345 

406 
467 
528 

588 
649 
709 

230 
291 
352 
412 
473 
534 

594 
655 
715 

236 
297 
358 

418 

479 
540 

600 
661 
721 

242 
303 
364 

425 
485 
546 

606 
667 
727 

733 

739 

745 

751 

757 

763 

769 

775 

78i 

788 

794 
854 
914 

^  974 
86  034 
094 

153 
213 

273 

800 
860 
920 

980 
040 

IOO 

159 
219 

279 
338" 

806 
866 
926 

986 
046 
106 

165 
225 
285 

812 
872 
932 

992 
052 

112 

171 
231 
291 

818 
878 
938 

998 
058 
118 

177 

237 
297 

824 
884 
944 
*(X>4 
064 
124 

183 

243 
303 

830 
890 
950 

*OIO 

070 
130 
189 

249 
308 

836 
896 
956 

*oi6 
076 
136 

195 

255 
3H 

842 
902 
962 

*O22 
082 
141 

201 
26l 
320 

848 
908 
968 

*028 

088 
147 
207 
267 
326 

332 

344 

404 

463 
522 

I81 
641 

700 

759 
817 
876 

350 

356 

362 

368 

374 

380 

386 

392 

45' 
510 

570 
629 
688 

747 
806 
864 

39o 
457 
516 

576 
635 
694 

753 
812 
870 

410 
469 
528 

587 
646 
705 

764 

823 
882 

415 
475 
534 

593 
652 
711 

770 
829 
888 

421 
481 
540 

599 
658 
717 

776 

835 
894 

427 
487 
546 

605 
664 
723 
782 
841 
goo 

433 
493 

552 

611 
670 
729 
788 

847 
906 

439 
499 
558 

617 
676 

735 

794 
853 
911 

445 
504 
564 

623 
682 
741 
800 

859 
917 

923 

929 

935 

941 

947 

953 

958 

964 

970 

976 

982 
87  040 
099 

I52 
216 

274 

332 
390 
448 

988 
046 
105 

163 

221 
280 

338 
396 

454 

994 
052 
in 

169 
227 
286 

344 

402 
460 

999 
058 
116 

'75 
233 
291 

349 
408 
466 

*005 

064 

122 

181 

239 
297 

355 
4i3 
47i 

*on 
070 
128 

186 

245 
303 
361 
419 
477 

*oi7 
075 
134 

192 
251 
309 

367 
425 
483 

*023 

08  1 

140 

198 

256 
315 

373 
489 

*O29 

087 
146 

204 
262 

320 

379 
437 
495 

*°35 
093 
151 

210 
268 
326 

384 
442 
500 

558 

9 

5o6 
o 

* 

518 
2 

523 
3 

529 
4 

535 
5 

54i 
6 

547 
7 

552 

8 

LOGARITHMS  OF  NUMBERS. 


N. 

>*—m—~~ 

750 

51 
52 
53 

54 
55 
56 

57 
58 
59 

7  CO 

61 
62 
63 

64 
65 
66 

67 
68 
69 

770 

71 
72 
73 

74 
75 
76 

77 
78 
79 

780 

81 
82 
83 

84 
85 
86 

87 

88 
89 

790 

91 
92 
93 

94 
95 
96 

97 
98 
99 

800 

O 

87  506 

1 
512 

2 

~ 

3 

523 

4 

529 

5 

^HMMMM 

535 

6 

MHMMM* 

541 

7 

«MMMM^ 

547 

8 

i^mmmtmm 

552 

9 

Iss" 

Pro 

MMMM 

1 

2 
3 
4 
5 
6 
7 
8 
9 

1 
2 
3 
4 
5 
6 
7 
8 
9 

p.  Pts. 

—  -^—  —  » 

0 

O.G 
1.2 
1.8 
2.4 
3.0 
3.6 
4.2 
4.8 
5.4 

5 

0.5 
1.0 
1.5 
2.0 
2.5 
3.0 
3.5 
4.0 
4.5 

622 
679 

737 
795 
852 

910 
967 
88  024 

570 
628 
685 

743 
800 
858 

915 
973 
030 

576 

633 
691 

749 
806 
864 

921 
978 
036 

|8i 
639 
697 

754 
812 
869 

927 
984 
041 

587 
645 
703 

760 
818 
875 

933 
990 
047 

593 

65i 
708 

766 

823 

881 

938 
996 

053 

599 
656 

714 

772 
829 
887 

944 

*OOI 

058 

604 
662 
720 

777 
835 
892 

950 
*oo7 
064 

610 
668 
726 

783 
841 
898 

*955 
*oi3 

070 

616 
674 
73i 

789 
846 
904 

961 
*oi8 
076 

081 

087 

093 

098 

104 

1  10 

116 

121 

127 

133 

138 

195 
252 

309 
366 

423 
480 
536 
593 

144 
20  1 
258 

315 
372 
429 

485 
542 
598 

150 

207 
264 

321 
377 
434 

491 

% 

156 
213 
270 

326 

383 
440 

497 
553 
610 

161 
218 
275 
332 

389 
446 

502 

559 
615 

167 
224 
281 

338 
395 
45i 
508 
564 
621 

173 
230 
287 

343 
400 

457 

513 
570 
627 

173 

235 
292 

349 
406 
463 

519 
576 
632 

184 
241 
298 

355 
412 
468 

525 
581 
638 

190 
247 
3«>4 
360 
417 
474 

530 

I87 
643 

649 

655 

660 

666 

672 

677 

683 

689 

694 

700 

70S 
III 
874 

89042 
098 
154 

711 
767 
824 

880 
936 
992 

048 
104 
159 

717 

773 
829 

885 
941 
997 

053 
109 
165 

722 
779 
835 
891 

947 
*oo3 

059 

H5 
170 

728 
784- 
840 

897 
953 
*oc9 

064 

120 

176 

734 
79° 
846 

902 

958 
*oi4 

070 
126 
182 

739 
795 
852 

£8 
964 

*020 
076 

I3I 

745 
801 

857 

9I3 
969 

*025 

08  1 
137 
193 

750 
807 
863 

919 

*975 
*03i 

087 

$ 

756 
812 
868 

925 

981 
*037 

092 
148 
204 

209 

215 

221 

226 

232 

237 

243 

248 

254 

260 

265 

% 
» 

542 
597 

$ 

271 
326 
382 

437 
492 
548 

603 
658 
713 

276 

$ 

$ 

553 
609 
664 
719 

282 
337 
393 

448 
504 

559 

614 
669 
724 

287 

$ 

454 
509 
564 

620 

675 
730 

$ 

404 

459 
5i5 
570 

625 
680 
735 

298 

354 
409 

465 
520 

575 

686 
741 

304 
360 

415 

470 

526 
581 

636 

691 

746 

310 

365 
421 

476 
53i 
5b<3 

642 
697 
752 

807 

315 

Z7\ 

426 

481 
537 
592 

647 
702 
757 

763 

768 

774 

779 

785 

790 

796 

801 

812 

818 

873 
927 

982 
90037 
091 

146 

200 

253 

823 
878 
933 
988 
042 
097 

'*! 
206 

260 

829 
883 
938 

993 
048 
1  02 

'.57 

211 
266 

$ 

944 
998 

33 

162 
217 
271 

840 
894 
949 
*004 
059 
"3 
168 

222 
276 

845 
900 

955 
*oo9 
064 
119 

173 
227 
282 

851 
905 
960 

*oi"5 
069 
124 

179 

233 
287 

856 
911 
966 

*O2O 
075 
129 

I84 
238 
293 

862 
916 
971 

*026 

080 
135 
189 

244 

298 

867 
922 
977 
*03i 
086 
140 

195 
249 
304 

309 

314 

320 

325 

331 

336 

342 

347 

352 

358 

N. 

0 

i 

2 

3 

4 

5 

<*   7   8 

0 

Prop.  Pts. 

1 6 


TABLE  I. 


N. 

0 

1 

2 

3 

4 

5 

0 

7 

8 

9 

~ 

Prop.  Pts. 

800 

90  309 

3H 

320 

325 

331 

~ 

342 

347 

352 

• 

01 

363 

369 

374 

380 

385 

390 

396 

401 

407 

412 

02 

417 

423 

428 

434 

439 

445 

450 

455 

461 

466 

03 

472 

477 

482 

488 

493 

499 

504 

509 

515 

520 

04 

526 

53I 

536 

542 

547 

553 

558 

563 

569 

574 

05 

580 

58? 

590 

596 

601 

607 

612 

617 

623 

628 

00 

634 

639 

644 

650 

655 

660 

666 

671 

677 

682 

• 

07 

687 

693 

698 

703 

709 

714 

720 

725 

730 

736 

08 

74  1 

747 

752 

757 

763 

768 

773 

779 

784 

789 

09 

795 

800 

806 

Sii 

816 

822 

827 

832 

838 

843 

810 

849. 

854 

~8~59~ 

865 

870 

875 

88  1 

886 

891 

897 

11 
12 

902 
956 

907 
961 

966 

918 
972 

924 
977 

929 

982 

934 
988 

940 
993 

998 

950 

*oo4 

1 

0.6 

13 

91  009 

014 

020 

025 

030 

036 

041 

046 

052 

057 

2 

1.2 

14 

062 

068 

073 

078 

084 

089 

094 

IOO 

105 

no 

3 

1.8 

2  A 

15 

116 

121 

126 

132 

137 

142 

148 

153 

158 

164 

,*t 
3  A 

16 

169 

174 

180 

185 

190 

196 

20  1 

206 

212 

217 

6 

»u 
3.6 

17 

222 

228 

233 

238 

243 

249 

254 

259 

265 

270 

7 

4.2 

18 

275 

281 

286 

291 

297 

302 

307 

312 

318 

323 

8 

4.8 

19 

328 

334 

339 

344 

350 

355 

360 

365 

371 

376 

9 

5.4 

820 

381 

387 

392 

397 

403 

408 

413 

418 

424 

429 

21 

434 

440 

445 

450 

455 

461 

466 

47  i 

477 

482 

22 

487 

492 

498 

503 

508 

5H 

519 

524 

529 

535 

23 

540 

545 

556 

561 

566 

572 

577 

582 

587 

24 

593 

598 

603 

609 

614 

619 

624 

630 

635 

640 

25 

645 

651 

656 

661 

666 

672 

677 

682 

687 

693 

26 

f  698 

703 

709 

7H 

719 

724 

730 

735 

740 

745 

27 

751 

756 

761 

766 

772 

777 

782 

787 

793 

798 

28 

803 

808 

814 

819 

824 

829 

834 

840 

845 

850 

29 

855 

861 

866 

871 

876 

882 

887 

892 

897 

9°3 

830 

908 

913 

918 

924 

929 

934 

939 

944 

950 

955 

31 

960 

965 

971 

976 

981 

986 

991 

997 

*OO2 

*oo7 

32 

02  OI2 

018 

023 

028 

033 

038 

044 

049 

054 

059 

1 

0.5 

33 

065 

070 

075 

080 

085 

091 

096 

101 

106 

in 

2 

1.0 

34 

117 

122 

127 

132 

137 

14^ 

148 

153 

158 

163 

3 

1.5 
2i\ 

35 
36 

I69 
221 

174 
226 

179 
231 

184 
236 

189 
241 

19? 
247 

200 
252 

205 
257 

2IO 
262 

215 
267 

5 
6 

.V 

2.5 
3.0 

37 

273  :  278 

283 

288 

293 

298 

304 

309 

314 

319 

7 

3.5   ' 

38 

324   330 

335 

340 

345 

350 

355 

361 

366 

371 

8 

4.0   1 

39 

376 

381 

3^7 

392 

397 

402 

407 

412 

4l8 

423 

0 

4.5 

840 

428 

433 

438 

443 

449 

454 

459 

464 

469 

474 

41 

480 

485 

490 

495 

500 

505 

511 

516 

521 

526 

42 

531 

536 

542 

547 

557 

562 

567 

572 

578 

43 

588 

593 

598 

603 

609 

614 

619 

624 

629 

44 

634 

639 

645 

650 

655 

660 

665 

670 

675 

68  1 

45 
46 

686 
737 

691 

742 

747 

701 
752 

706 
758 

711 

763 

716 
768 

722 
773 

727 

778 

732 
783 

47 
48 

788 
840 

793 

845 

799 
850 

804 
85? 

809 
860 

814 

865 

819 
870 

824 
875 

829 
881 

834 
8S6 

49 

891 

896 

901 

906 

911 

916 

921 

927 

932 

937 

850 

942 

947 

952 

957 

962 

967 

973 

978 

983 

988 

N. 

O 

1 

MBMMIM 

3 

••••MMM 

4 

5 

6 

7 

8 

9 

Prop.Pts. 

LOGARITHMS  OF  NUMBERS. 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Pro] 

).  PtS. 

850 

92  942 

947 

952 

957 

962 

967 

973 

978 

983 

988 

51 
52 
53 

54 
i  55 
50 

57 
58 
59 

993 
93  044 
095 

146 
197 
247 

298 
349 
399 

998 
049 

IOO 

151 

202 
252 

303 

354 
404 

*oo3 
054 
105 

156 
207 
258 

308 

359 
409 

*oo8 
059 
no 

161 

212 
263 

3<3 

364 

414 

*OI3 

064 
"5 
166 
217 
268 

3i8 
369 
420 

*oi8 
069 

120 

171 
222 
273 
323 

374 
425 

*024 

075 
125 

176 

227 

278 
328 

379 
430 

*029 

080 
131 

181 
232 
283 

334 
384 
435 

*Q34 
085 
136 

1  86 

237 
288 

339 
389 
440 

*Q39 
090 
141 

192 

242 
293 

544 
394 
445 

1 
2 
3 

0 

0.6 
1.2 

1.8 
24. 

860 

450 

455 

460 

465 

470 

475 

480 

485 

490 

495 

5 

3.0 

61 
62 
63 

64 
65 
66 

67 
68 
60 

500 

551 
601 

651 
702 
752 

802 
852 
902 

505 

656 
707 
757 
807 

857 
907 

Sx° 
561 

611 

661 
712 
762 

812 
862 
912 

& 

616 

666 
717 
767 

817 
867 
917 

520 

57i 
621 

671 
722 
772 

822 
872 
922 

52^ 
576 

626 

676 
727 
777 
827 
877 
927 

531 
|8i 
631 

682 
732 
782 

832 
882 
932 

536 
586 
636 

687 
737 
787 

837 
887 
937 

541 
59i 
641 

692 
742 
792 

842 
892 
942 

546 
596 
646 

697 
747 
797 
847 
897 
947 

6 
7 
8 
9 

3.6 
4.2 
4.8 
5.4 

870 

952 

957 

962 

967 

972 

977 

982 

987 

992 

997 

6' 

71 
72 
73 

74 
75 

76 

77 
78 
79 

94  002 
052 

101 

151 

201 
250 

300 

349 
399 

007 

°52 
106 

156 
206 
255 

305 
354 
404 

012 
062 
III 

161 

211 
260 

310 

359 

409 

017 

067 

116 

166 
216 

265 

315 

364 
4H 

022 
072 
121 

171 
221 
270 

320 

369 
419 

027 

077 
126 

176 
226 
275 

325 
374 
424 

032 
082 
I3i 
181 

280 

330 

379 
429 

037 
086 

136 

1  86 
236 
285 

335 
384 
433 

042 
091 
141 

191 
240 
290 

340 
389 
438 

047 
096 
146 

196 

245 
295 

345 
394 
443 

1 
2 
3 
4 
5 
6 
7 
8 
9 

0.5 
1.0 
1.5 
2.0 
2.5 
3.0 
3.5 
4.0 
4.5 

880 

448 

453 

458 

463 

468 

473 

478 

483 

488 

493 

81 
82 
83 

84 
85 
86 

87 
88 
89 

498 

547 
596 

645 
694 
743 

792 
841 
890 

503 

I52 
601 

650 
699 
748 

797 
846 

895 

507 

557 
606 

655 
704 

753 
802 
851 
900 

512 
562 

660 
709 
758 

807 
856 
905 

517 

567 

616 

665 
7H 
763 

812 
86  1 
910 

522 

571 
621 

670 
719 
768 

817 
866 
915 

522 

576 
626 

675 

724 

773 
822 
871 
919 

532 

630 

680 
729 
778 

827 
876 
924 

537 
586 

635 
685 
734 
783 

832 
880 
929 

542 

I91 

640 

689 
738 
787 

836 
885 
934 

1 
2 
3 

4 

0.4 
0.8 
1.2 

890 

939 

944 

949 

954 

959 

963 

968 

973 

978 

983 

4 
5 

1.6 
2  0 

91 
92 
93 

94 

95 
96 

97 

98 
99 

988 
95  036 
085 

134 
182 
231 

279 
328 
376 

993 
041 
090 

139 
187 
236 

284 
332 
38i 

998 
046 
095 

143 
192 
240 

289 

& 

*002 
051 
IOO 

148 
197 
245 

294 
342 
390 

*<x>7 
056 
105 

153 

202 
250 

209 

347 
395 

*OI2 

061 
109 

158 
207 

255 

303 
352 
400 

*oi7 
066 
114 

163 

211 

260 

308 

357 
405 

*022 
071 
119 

1  68 
216 
265 

3<3 
361 

410 

*027 

075 
124 

173 

221 
270 

318 
366 
415 

*O32 

080 
129 

177 
226 
274 

323 
371 
419 

6 
7 
8 
9 

2.4 
2.8 
3.2 
3.6 

900 

424 

429 

434 

439 

444 

448 

453 

458 

463 

468 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Pro 

p.  Pts. 

i8 


TABLE  I. 


N. 

O 

1   2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Ptg. 

900 

95  424 

429  434 

439 

444 

448 

453 

"458" 

"463" 

"468" 

01 

472 

477 

482 

487 

492 

497 

501 

506 

5" 

516 

02 
03 

521 
569 

525 
574 

530 
578 

535 

540 
588 

545 
593 

550 
598 

602 

559 
607 

564 
612 

04 

617 

622 

626 

631 

636 

641 

646 

650 

655 

660 

05 

665 

670 

674 

679 

684 

689 

694 

698 

703 

708 

06 

713 

718 

722 

727 

732 

737 

742 

746 

756 

07 

761 

766 

770 

775 

780 

785 

789 

794 

799 

804 

08 

809 

813 

818 

823 

828 

o 

837 

842 

847 

852 

09 

856 

861 

866 

871 

875 

880 

885 

890 

895 

899 

910 

904 

909 

914 

918 

923 

928 

933 

938 

942 

947 

11 

952 

957 

*961 

966 

*971 

976 

980 

985 

990 

995 

12 

999 

*oo4 

*oi4 

*023 

*028 

*O33 

*o38 

*042 

1 

0.5 

13 

96047 

052 

057 

061 

066 

071 

076 

080 

085 

090 

2 

1.0 

14 
15 

095 
142 

099 
147 

104 
152 

109 
156 

114 
161 

118 
166 

123 

128 
175 

III 

137 

185 

3 

4 
p. 

1.5 
2.0 

9  f\ 

16 

190 

194 

199 

204 

209 

213 

218 

223 

227 

232 

o 
6 

£  .O 

3.0 

17 
18 

as 

242 
289 

246 
294 

251 
298 

256 
303 

261 
308 

265 
313 

270 
317 

275 
322 

280 
327 

7 

8 

3.5 
4.0 

19 

332 

336 

341 

346 

350 

355 

360 

365 

369 

374 

9 

4.5 

920 

379 

384 

388 

393 

398 

402 

407 

412 

417 

421 

21 

426 

431 

435 

440 

445 

450 

454 

459 

464 

468 

22 

473 

478 

483 

487 

492 

497 

501 

506 

5" 

515 

23 

520 

525 

530 

534 

539 

544 

548 

553 

558 

562 

24 
25 

£4 

572 
619 

577 
624 

581 
628 

586 
633 

591 
638 

I95 
642 

600 
647 

605 
652 

609 
656 

26 

661 

666 

670 

675 

680 

685 

689 

694 

699 

703 

27 
28 

708 

713 
759 

717 
764 

722 
769 

727 
774 

778 

736 
783 

788 

745 
792 

750 

29 

802 

806 

811 

816 

820 

825 

830 

834 

839 

844 

930 

848 

853 

858 

862 

867 

872 

876 

881 

886 

890 

31 
32 

895 
942 

900 
946 

904 

909 
956 

914 
#96o 

918 
965 

923 
970 

928 
974 

932 
979 

984 

1 

0.4 

33 

988 

993 

997 

*OO2 

*OII 

*oi6 

*02I 

*O2S 

*O3O 

2 

0.8 

34 

97  035 

039 

044 

049 

053 

058 

063 

067 

072 

077 

3 

1.2 

10 

35 

08  1 

086  1  090 

095 

100 

104 

109 

114 

118 

123 

.0 

2(1 

36 

128 

132  1  137 

142 

146 

155 

1  60 

165 

169 

6 

.V 

2.4 

37 

174 

179 

183 

188 

192 

197 

202 

206 

211 

216 

7 

2.8 

38 

220 

225 

230 

234 

239 

243 

248 

253 

257 

262 

8 

3.2 

39 

267 

271 

276 

280 

285 

290 

294 

299 

304 

308 

9 

3.6 

940 

313 

317 

322 

327 

331 

336 

340 

345 

350 

354 

41 

359 

364 

368 

373 

377 

382 

387 

391 

396 

400 

42 

405 

410 

414 

419 

424 

428 

433 

437 

442 

447 

43 

456 

460 

465 

470 

474 

479 

483 

488 

493 

44 

497 

502 

!  506 

5" 

516 

520 

525 

529 

534 

539 

45 

543 

548 

552 

557 

562 

566 

575 

580 

585 

46 

589 

594 

598 

603 

607 

612 

617 

621 

626 

630 

47 

633 

640 

644 

649 

653 

658 

663 

667 

672 

676 

48 

68  1 

685 

690 

695 

699 

704 

708 

713 

717 

722 

49 

727 

736 

740 

745 

749 

754 

759 

763 

768 

950 

772 

777 

782 

786 

791 

795 

800 

804 

809 

813 

N. 

O 

1  |  2 

3 

4 

5 

6 

r 

8 

9 

Prop.  Pte. 

LOGARITHMS  OF  NUMBERS. 


N. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.Pte.|| 

950 

97  772 

777 

782 

786 

791 

795 

800 

804 

809 

813 

51 

818 

823 

827 

832 

836 

841 

845 

850 

855 

859 

52 

864 

868 

873 

877 

882 

886 

891 

896 

900 

90S 

53 

909 

914 

918 

923 

928 

932 

937 

941 

946 

950 

54 

955 

959 

964 

968 

973 

978 

982 

987 

991 

996 

55 

98  ooo 

00$ 

009 

014 

019 

023 

028 

032 

037 

041 

56 

046 

050 

055 

059 

064 

068 

073 

078 

082 

087 

' 

57 

58 

091 
137 

096 
141 

100 

146 

105 
150 

!SS 

114 

159 

118 
164 

III 

127 
173 

132 
177 

59 

182 

186 

191 

195 

200 

204 

209 

214 

218 

223 

960 

227 

232 

236 

241 

245 

250 

254 

259 

263 

268 

• 

61 

272 

277 

281 

286 

290 

295 

299 

304 

308 

313 

6 

62 

318 

322 

327 

331 

336 

340 

345 

349 

354 

358 

1 

0.5 

63 

363 

367 

372 

376 

381 

385 

390 

394 

399 

403 

2 

1.0 

64 
65 

408 

412 
457 

4J7 

462 

421 
466 

426 
47i 

430 
475 

480 

439 
484 

444 
489 

448 
493 

3 

4 

1.5 
2.0 
2  ft 

66 

498 

502 

507 

5" 

516 

520 

525 

529 

534 

538 

6 

3.0 

67 

68 
69 

F 

632 

547 
592 
637 

552 
597 
641 

I*6 
601 

646 

I? 
605 

650 

565 
610 

655 

570 
614 
659 

574 
619 
664 

579 
623 
668 

673 

7 
8 
9 

3.& 
4.0 
4.6 

970 

677 

682 

686 

691 

695 

700 

704 

709 

713 

717 

71 

722 

726 

731 

735 

740 

744 

749 

753 

758 

762 

1 

72 

767 

771 

776 

780 

784 

789 

793 

798 

802 

807 

1 

73 

fell 

816 

820 

825 

829 

834 

838 

843 

847 

851 

1 

74 

856 

860 

865 

869 

874 

878 

883 

887 

892 

896 

| 

75 

900 

9^5 

909 

914 

918 

923 

927 

932 

936 

941 

I 

76 

945 

949 

954 

958 

963 

967 

972 

976 

981 

985 

1 

77 

989 

994 

098 

"003 

*oo7 

*OI2 

*oi6 

*O2I 

*025 

*029 

73 
79 

99034 
078 

038 
083 

043 
087 

047 
092 

052 
096 

056 
100 

061 
105 

065 
109 

069 
114 

074 

118 

I 

980 

123 

127 

131 

136 

140 

145 

149 

154 

158 

162 

81 

167 

171 

176 

1  80 

185 

189 

193 

I98 

202 

207 

II 

82 

211 

216 

220 

224 

229 

233 

238 

242 

247 

251 

1 

0.4 

83 

255 

260 

264 

269 

273 

277 

282 

286 

291 

295 

2 

0.8 

84 

300 

304 

308 

313 

317 

322 

326 

330 

335 

339 

3 

1.2 

1  ct 

85 

344 

348 

352 

357 

361 

366 

370 

374 

379 

383 

9  n 

86 

388 

392 

396 

401 

405 

410 

414 

419 

423 

427 

6 

24 

87 

432 

436 

441 

445 

449 

454 

458 

463 

467 

471 

7 

2.8 

88 

476 

480 

484 

489 

493 

498 

502 

506 

511 

515 

8 

3.2 

89 

520 

524 

528 

533 

537 

542 

546 

550 

555 

559 

9 

3.6 

990 

564 

568 

572 

577 

581 

585 

590 

594 

599 

603 

91 

607 

612 

616 

621 

62$ 

629 

634 

638 

642 

647 

1 

92 

651 

656 

660 

664 

669 

673 

677 

-682 

686 

691 

93 

695 

699 

704 

708 

712 

717 

721 

726 

730 

734 

94 

739 

743 

747 

752 

756 

760 

765 

769 

774 

778 

95 

782 

787 

791 

795 

800 

804 

808 

813 

8i7 

822 

1 

96 

826 

830 

835 

839 

843 

848 

852 

856 

861 

865 

1 

97 

870 

874 

878 

883 

887 

801  896 

000 

904 

909 

1  98 
99 

913 
957 

^61 

922 
965 

926 
970 

930 
974 

935 
978 

939 
983 

944 
987 

94** 
991 

952 
996 

1000 

00  000 

004 

009 

013 

017 

022 

026 

030 

035 

039 

9 

N. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.Pta.ll 

|l 

2O 


TABLE  I. 


N. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

1000 

ooo  ooo 

043 

087 

130 

174 

217 

260 

304 

347 

391 

1001 

434 

477 

521 

564 

608 

651 

694 

738 

78i 

824 

1002 

868 

911 

954 

998 

*O4i 

*o84 

*I28 

*I7I 

*2I4 

*258 

1003 

ooi  301 

344 

388 

43i 

474 

517 

561 

604 

647 

690 

44 

1004 

734 

777 

820 

863 

907 

950 

993 

*036 

*o8o 

*I23 

1 

4.4 

1005 

002  1  66 

209 

252 

296 

339 

425 

468 

512 

555 

2 

8.8 

1006 

598 

641 

684 

727 

771 

814 

857 

900 

943 

986 

3 

13.2 

1007 

003  029 

073 

116 

159 

202 

245 

288 

331 

374 

417 

4 

17.6 

1008 

461 

504 

547 

590 

633 

676 

719 

762 

805 

848 

5 

22.0 

1009 

891 

934 

977 

*020 

*o63 

*io6 

*i49 

*I92 

*235 

*278 

6 

IT 

26.4 

OA  Q 

1010 

004  321 

364 

407 

450 

493 

536 

579 

622 

665 

708 

f 

8 

oU.  o 

35.2 

1011 

75i 

794 

837 

880 

923 

966 

*oo9 

"052 

*095 

*I38 

9 

39.6 

1012 

005  i  80 

223 

266 

309 

352 

395 

438 

481 

524 

567 

1013 

609 

652 

695 

738 

781 

824 

867 

909 

952 

995 

1014 

006  038 

08  1 

124 

166 

209 

252 

295 

338 

380 

423 

1015 

466 

509 

552 

594 

637 

680 

723 

765 

808 

851 

43 

1016 

894 

936 

979 

*022 

*o6s 

*io7 

*i5o 

*I93 

'236 

*278 

1 

4.3 

1017 

007  321 

364 

406 

449 

492 

534 

577 

620 

-662 

705 

2 

8.6 

1018 

748 

790 

833 

876 

918 

961 

*oo4 

*046 

*o89 

*I32 

3 

12.9 

1019 

008  174 

217 

259 

302 

345 

387 

430 

472 

515 

558 

4 

17.2 

O1  K 

1020 

600 

643 

_6_8s_ 

728 

770 

813 

856 

898 

941 

983 

6 

21.  o 
25.8 

1021 

009  026 

068 

in 

153 

196 

238 

281 

323 

366 

408 

7 

30.1 

1022 

45i 

493 

536 

578 

621 

f\f\^ 

706 

748 

791 

833 

8 

34.4 

1023 

876 

918 

961 

*oo3 

*°45 

*o88 

*I30 

*I73 

*2I5 

*258 

9 

38.7 

1024 

oio  300 

342 

385 

427 

470 

512 

554 

597 

639 

681 

1025 

724 

766 

809 

851 

893 

936 

978 

*020 

*o63 

*ios 

1026 

on  147 

190 

232 

274 

317 

359 

401 

444 

486 

528 

1027 

570 

613 

655 

697 

740 

782 

824 

866 

909 

95i 

42 

1028 

993 

*035 

*o78 

*I20 

*I62 

*2O4 

*247 

*289 

*33i 

*373 

1 

4/2 

1029 

012  415 

458 

500 

542 

584 

626 

669 

711 

753 

795 

2 

8.4 

1030 

837 

879 

922 

964 

*oo6 

*048 

*O9O 

*I32 

*I74 

*2I7 

8 

12.6 

1031 

013  259 

301 

343 

385 

427 

469 

5ii 

553 

596 

638 

4 

16.8 

91  A 

1032 

680 

722 

764 

806 

848 

890 

932 

974 

*oi6 

="058 

XI  .  v 

OK  O 

1033 

014  100 

142 

184 

226 

268 

310 

352 

395 

437 

479 

7 

4t).  £i 

29.4 

1034 

521 

563 

605 

647 

689 

730 

772 

814 

856 

898 

8 

33.6 

1035 

940 

982 

*024 

*o66 

*io8 

*i5o 

*I92 

*234 

*276 

*3i8 

9 

37.8 

1036 

015  360 

402 

444 

485 

527 

569 

611 

653 

695 

737 

1037 

779 

821 

863 

904 

946 

988 

*O3O 

*O72 

*ii4 

*I56 

1038 

016  197 

239 

281 

323 

365 

407 

448 

490 

532 

574 

1039 

616 

657 

699 

74i 

783 

824 

866 

908 

950 

992 

41 

1040 

017  033 

075 

117 

159 

200 

242 

284 

326 

367 

409 

1 

4.1 

1041 

45i 

492 

534 

576 

618 

659 

701 

743 

784 

826 

2 

8.2 

10i2 

868 

909 

95i 

993 

*034 

*076 

*n8 

*I59 

*20I 

*243 

3 

12.3 

1043 

018  284 

326 

368 

409 

451 

492 

534 

576 

6I7 

659 

4 
5 

16.4 
20.5 

1044 

700 

742 

784 

825 

867 

908 

950 

992 

*033 

*075 

6 

24.6 

1045 

019  116 

158 

199 

241 

282 

324 

366 

407 

449 

490 

7 

28.7 

1046 

532 

573 

615 

656 

698 

739 

781 

822 

864 

905 

8 

32.8 

1047 

947 

988 

*030 

*07I 

*ii3 

*I54 

*I95 

*237 

*278 

*320 

9 

36.9 

1048 
1049 

020  361 
775 

403 
817 

444 
858 

486 
900 

527 
94i 

568 
982 

610 

*O24 

651 
"065 

693 
*io7 

734 
*I48 

1050 

021  189 

231 

272 

313 

355 

396 

437 

479 

520 

561 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

LOGARITHMS  OF  NUMBERS. 


21 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

1050 

1051 
1052 
1053 

1054 
1055 
]056 

1057 
1058 
1059 

10CO 

1061 
1062 
1063 

1064 
1065 
1066 

1067 
1068 
1069 

1070 

1071 
1072 
1073 

1074 
1075 
1076 

1077 
1078 
1079 

1080 

1081 
1082 
1083 

1084 
1085 
1086 

1087 
1088 
1089 

1090 

1091 
1092 
1093 

1094 
1095 
1096 

1097 
1098 
1099 

1100 

— 

021  189 

231 

272 

3i3 

355 

396 

437 

479 

520 

561 

1 

2 
3 
4 
5 
6 
7 
8 
9 

1 
2 
3 
4 
5 
6 
7 
8 
9 

1 
2 
3 
4 
5 
6 
7 
8 
9 

i 

2 
3 

4 
5 
6 
7 
8 
9 

"FT 

42 

4.2 
8.4 
12.6 
16.8 
21.0 
25.2 
29.4 
33.6 
37.8 

41 

4.1 
8.2 
12.3 
16.4 
20.5 
24.6 
28.7 
32.8 
36.9 

40 

4.0 

8.0 
12.0 
16.0 
20.0 
24.0 
28.0 
32.0 
36.0 

89 

3.9 

7.8 
11.7 
15.6 
19.5 
23.4 
27.3 
31.2 
35.1 

M^M^MMMV 

op.  Pts. 

603 

022  Ol6 
42S 

841 
023  252 

664 

024  075 
486 
896 

644 
057 
470 

882 
294 
705 

116 

527 
937 

685 

5" 

923 

335 
746 

157 
568 
978 

727 
140 
552 
964 
376 
787 

198 
609 
*oi9 

768 
181 
593 
*oos 

417 
828 

239 
650 
*o6o 

809 

222 
635 

*047 
458 
870 

280 
691 

*IOI 

851 
263 
676 

*oS8 

499 
911 

321 
732 

*I42 

892 

3^5 
717 

*I29 
541 
952 

363 

773 
*i83 

933 
346 
758 

*I70 
582 
993 
404 
814 

*224 

974 
387 
799 

*2II 
623 
*034 

445 
855 

*265 

025  306 

347 

388 

429 

470 

SH 

552 

593 

634 

674 

,  715 
026  125 

533 
942 
027  350 
757 
028  164 

57i 
978 

029  384 

756 
165 
574 
982 
390 
798 

205 
612 
*oi8 

797 

20<$ 
615 
*023 

431 
839 

246 

653 

*059 

838 
247 
656 

"064 
472 
879 

287 
693 

*IOO 

879 
288 

697 
*io5 

5i3 
920 

327 
734 
*I4O 

920 
329 
737 

"146 

II? 

368 

775 
*i8i 

961 
370 
778 

*i86 
594 

*002 

409 

*8'5 
S^T)  j 

*002 

411 
819 

*227 

635 
*O42 

449 
856 

*262 

*043 

452 
860 

*268 
676 
*o83 

490 
896 
*3Q3 
708 

*o84 
492 
901 

*309 
716 

*I24 

53i 
937 
*343 

424 

465 

506 

546 

587 

627 

668 

749 

789 
030  195 
600 

031  004 
408 
812 

032  216 
619 

033  021 

830 

*35 
640 

045 
449 
853 
256 
659 
062 

871 
276 
681 

085 
489 
893 
296 
699 

102 

9l\ 
316 

721 

126 
530 

933 

337 
740 
142 

952 

357 
762 

166 
570 
974 

377 
780 
182 

77 

992 

397 
802 

206 
610 
*oi4 

417 

820 
223 

*Q33 
438 
843 

247 
651 
*054 

458 
860 
263 
665 

*073 
478 
883 

287 
691 
*°95 

498 
901 
303 

705" 

*H4 
519 
923 
328 

732 
*i35 

538 
941 
343 

*I54 
559 
964 

368 
772 
*i75 

'5Z8 
981 

384 

424 

464 

504 

544 

625 

745 

785 

826 
034  227 
628 

035  029 

43° 
830 

036  230 
629 
037  028 

866 
267 
669 

069 
470 
870 

269 

906 
308 
709 

109 

510 
9IO 

309 

709 

108 

946 
348 
749 
149 

550 
950 

349 
749 
148 

986 
388 
789 

190 

590 
990 

389 
789 
187 

*O27 

428 
829 

230 

630 

*O3O 

0 

227 

*o67 
468 
869 

270 
670 
*070 

469 
868 
267 

*io7 
508 
909 

310 
710 

*IIO 

509 
908 
307 

*I47 
548 
949 

350 

750 
*i5o 

549 
948 
347 

"187 
588 
989 

390 
790 
*i9o 

9 

387 

426 

466 

506 

546 

586 

984 
382 

779 
176 

573 
969 

365 
761 
*i$6 

626 

665 

705 

745 

*H3 
54i 
938 

335 
73i 

*I27 

523 
919 

*3U 
708 

8 

785 

825 
038  223 
620 

039  017 

414 
811 

040  207 
602 
998 

041  393 

—  B^-M^—  ^«B 

O 

865 
262 
660 

057 

454 
850 

246 
642 
*Q37 

904 
302 
700 

097 

493 
890 

286 
68  1 
*077 

914 
342 
739 
136 
533 
929 

325 
721 
*ii6 

*O24 

421 
819 

216 
612 
*oo9 

405 
800 
*'95 

"064 
461 
859 

?55 
652 

*o48 

444 
840 

I23i 
630 

6 

*io3 
501 

295 

484 
879 
*274 

*i83 
580 
978 

374 
771 
*i67 

563 
958 

*353 

432 

•MMMMW 

1 

472 
2 

5H 
3 

55i 
4 

590 
5 

669 

7 

748 
9 

TABLE  II. 


TABLE   II. 

CONSTANTS  WITH  THEIR  LOGARITHMS. 

Number. 

Logarithm. 

Ratio  of  circumference  to  diameter,  TV, 

3.14159265 

0.49714  99 

..      n\ 

9.86960440 

0.99429  97 

•  •                            .•                              ••        27T, 

6.28318531 

0.79817  99 

..      ^, 

I-77245385 

0.2485749 

Number  of  degrees  in  circumference, 

360° 

2-5563025 

•  •          minutes           •  • 

21600' 

4-33445  38 

seconds            «• 

1296000" 

6.11260  50 

Degrees  in  arc  equal  to  radius, 

57°-  2957795 

1.75812  26 

Minutes      ..      ..          •• 

3437'.  74677 

3-53627  39 

Seconds      ••      ••          •• 

2o6264//.8o6 

5.31442  51 

Length  of  arc  of  i  degree, 

.01745329 

8.24187  74—10 

..i  minute, 

.00029089 

6.46372  61—  10 

..            ..i  second, 

.000004848 

4.68557  49—10 

Number  of  hours  in  i  day, 

24 

1.38021   12 

•  •          minutes    •  • 

1440 

3.15836  25 

.  •         seconds      • 

86400 

4-9365«  37 

Number  of  days  in  Julian  year, 

365.25 

2.5625902 

Naperian  base, 

2.718281828 

0.43429  45 

Modulus  of  common  logarithms, 

0.434294482 

9.6377843—10 

Hours  in  which  earth  revolves  through 

arc  equal  to  radius, 

3.8197186 

0.58203  14 

Minutes  of  time        ..        ..         .. 

229.18312 

2.36018  26 

Seconds  of  time        

'3750-987 

4-1383339 

III— SINES  AND  TANGENTS  OF  SMALL  ANGLES.  25 


TABLE  III. 


FOR 

SINES  AND  TANGENTS  OF  SMALL  ANGLES. 

TO  FIND  THE  SINE  OB  TANGENT  I 

Log  sin  a  =  log  a  (in  seconds)  +  & 
Log  tan  a  =  log  a  (in  seconds)  +  T. 

TO  FIND  A  SHALL  ANGLE  FROM  ITS  SINE  OB  TANGENTs 

Log  a  (in  seconds)  =  log  sin  a  +  •?'• 
Log  a  (in  seconds)  as  log  tan  a  +  2*. 


26 


TABLE  III. 


0° 

99 

9 

L.  Sin. 

8 

T 

S' 

T' 

0 
60 
1  2O 

1  80 
240 

0 

I 
2 

3 
4 

6.4^73 
•  76476 

7.06579 

4-68557 
.68557 
.68557 
.68557 
.68557 

4.68557 
.68557 
.68557 
.68557 
.68558 

S-3I443 
.31443 
.31443 
.31443 
.31443 

5-31443 
.31443 
.31443 
.31443 
.31442 

300 
360 
420 
480 
540 

1 

I 

9 

7.16270 
.24188 
.30882 
.36682 
.41797 

4.68557 
.68557 
.68557 
.68557 
.68557 

4.68558 
.68558 
.68558 
.68558 
.68558 

5-31443 
31443 
.31443 
•31443 
.31443 

5-31442 
•3I442 
•31442 
.31442 
.31442 

600 
660 
720 
780 
840 

10 
ii 

12 
13 

H 

7.46373 
.50512" 
•54291 

n 

-4:68557 
.68557 
•68557 
.68557 
•68557 

4-68558 
.68558 
.68558 
.68558 
.68558 

S.3I443 
.31443 
.31443 
•31443 
.31443 

5-3I442 
.31442 
.31442 
•3I442 
.  3*442 

s 

960 

I02O 
1080 
1140 

!! 
[I 

19 

7:S& 

.69417 

.71900 
.74248 

4.6«557 
.68557 
.68557 
.68557 
.68557 

4.68558 
.68558 
.68558 
.68558 
.68558 

5-31443 
.31443 
.31443 
.31443 
.31443 

5-31442 
.31442 
.3H42 
.3*442 
.3*442 

1200 
1260 
1320 
1380 
1440 

20 
21 
22 
23 

24 

7.76475 
.78594 
.80615 

.82545 
.84393 

4.68557 
.68557 
.68557 
.68557 
.68557 

4-68558 
.68558 
.68558 
.68558 
.68558 

5-3I443 
.31443 
.31443 
.31443 
.31443 

5-31442 
•3I442 
.31442 
..31442 
.31442 

1500 
1560 

1620 
1680 
1740 

3 

% 

29 

7.86166 
.87870 
.89509 
.91088 
.92612 

4.68557 
.68557 
.68557 
.68557 
.68557 

4-68558 
.68558 
.68558 
.68558 
.68559 

5-31443 
.31443 
.31443 
.31443 
•31443 

5.31442 
.31442 
.31442 
•3I442 
.3*44* 

1800 

1860 
1920 
1980 
2040 

30 
3i 
32 
33 
34 

7.94084 
.95508 
.96887 
.98223 
•99520 

4-68557 
.68557 
.68557 
.68557 
.68557 

4-68559 
.68559 
.68559 
.68559 
.68559 

5-31443 
.31443 
.31443 
•3I443 
•3I443 

5.3I44I 
•3I44i 
•3*441 
.3*441 
•  3*44* 

SHOO 

2l6o 
2220 
2280 
2340 

8 
8 

39 

8.00779 
.02002 
.03192 

•04350 
•05478 

4.68557 
•68557 
.68557 
.68557 
.68557 

4-68559 
.68559 
.68559 
.68559 
.68559 

5.3I443 
.31443 
.31443 
.31443 
.31443 

5.3*44* 
•3*441 
.3*44* 
.31441 
.31441 

2400 
2460 
2520 
2580 
2640 

40 

4i 
42 

43 

44 

8.06578 
.07650 

.09718 
.10717 

4'S551 
.68556 

.68556 
.68556 
.68556 

4-68559 
.68560 
68560 
.68560 
.68560 

5-31443 
•3I444 
•3I444 
.31444 
.31444 

5-3*44* 
.3*440 
.31440 
.3*440 
•  3*440 

2700 
2760 
2820 
2880 
2940 

9 

47 
48 

49 

8.11693 
.12647 
.13581 
.14495 
.15391 

4.68556 
.68556 
.68556 
.68556 
.68556 

4.68560 
.68560 
.68560 
.68560 
.68560 

5-3I444 
•3I444 
.31444 
.3!444 
.  3*444 

5.3I440 
.3*440 
.31440 
•3I440 
.3*440 

3000 
3060 
3120 
3180 
3240 

50 
51 
52 
S3 

54 

8.16268 
.17128 

-I797I 
.18798 
.  19610 

4-68556 
.68556 
.68556 
.68556 
.68556 

4.68561 
.68561 
.68561 
.68561 
.68561 

5-31444 
•3I444 
•3M44 
.51444 
•3I444 

5.31439 
.31439 
.3H39 
.31439 
.3H39 

3300 
3360 
3420 
3480 

3540 

P 
P 

59 

8.20407 
.21189 
.21958 
.22713 

•23456 

4-68556 
.68556 
.68555 
.68555 
.68555 

4.68561 
.68561 
.68561 
.68562 
.68562 

5-31444 
•3I444 
•3I445 
.31445 
•31445 

5-3I439 
.3H39 
.31439 
.3H38 
.31438 

3600 

60 

8.24186 

4-68555 

4.68562 

5-3I445 

5  3H38 

SINES  AND  TANGENTS  OF  SMALL  ANGLES. 


1° 

M 

t 

L.Sin. 

8 

T 

8' 

r 

3600 

3660 

3720 
3780 
3840 

o 
I 

2 

3 

4 

8.24186 
.24907 
.25609 
.26304 
.26988 

4-68555 
.68555 

.68555 
.68555 

.68555 

4.68562 
.68562 
.68562 
.68562 
.68563 

5-3I445 
.31445 
31445 
.31445 
.31445 

5.31438 
•31438 
.31435 
.31438 

•3H37     i 

3900 
3960 
4020 
4080 
4140 

i 
I 

9 

8.27661 
.28324 

.28977 
.29621 

.30255 

4-68555 
.68555 

.68555 
.68555 
.68555 

4.68563 
.68563 
.68563 
.68563 
.68563 

5-31445 
.31445 
.31445 
•31445 
.31445 

5-3H37     1 
.3H37 
.3H37 
.3H37 
•31437 

4200 
4260 
4320 

4440 

10 
ii 

12 
13 

H 

8.30879 

.3H95 
.32103 
.32702 
.33292 

4-68554 
.68554 
-68554 
.68554 
.68554 

4-68563 
.68564 
.68564 
.68564 
.68564 

5-31446 
.31446 
.31446 
.31446 
.31446 

5.3H37 

.31436 
.31436 

4620 
4680 
4740 

% 

19 

8.33875 
•34450 
.35018 
.35578 
.36131 

4-68554 
.68554 
.68554 
.68554 
.68554 

4.68564 
.68565 
.68565 
.68565 
.68565 

5-31446 
.31446 
.31446 
.31446 
.31446 

5.3I436 
.3H3S 
.31435 
.31435 
.31435 

4800 
4860 
4920 
498o 

5040 

20 

21 
22 
23 

24 

8.36678 
.37217 

'38276 
.38796 

4.68554 
.6S553 
.68553 
.68553 

.68553 

4.68565 
.68566 
.68566 
.68566 
.68566 

5-  3M46 
.31447 
.31447 
.31447 
•  3  H47 

5.31435 
.31434 
.31434 
.3H34 
.31434 

5100 
5160 
5220 
5280 
5340 

11 
11 

29 

'40320 
.40816 

4-68553 
.68553 
.68553 
.68553 
.68553 

4.68566 
.68567 
.68567 
.68567 
.68567 

5-3I447 
.31447 
.31447 
.31447 
•31447 

5-3I434 

^433 
.31433 
.31433 

5400 

5520 
558o 
5640 

30 
31 
32 

33 
34 

8  41792 
.42272 
42746 
.43216 
.43680 

4-68553 
.68552 
.68552 
.68552 
.68552 

4-68567 
.6856$    • 
.68568 
.68568 
.68568 

5-31447 
.31448 
.31448 
.31448 
•31448 

5-3H33 
.3H32 

'3H32 

5700 
5760 
5820 
5880 

CO  CO  CO  CO  CO 

8.44139 

.44594 
.45044 

.45489 
.45930 

4-68552 
.68552 
.68552 
.68552 
.68551 

4-68569 
.68569 
.68569 
.68569 
.68569 

5-3I448 
.31448 
.31448 
.31448 
.31449 

^31431 
•3H3I 
.3H3I 
•3I43I 

6060 
6120 
6180 
6240 

40 
41 
42 
43 
44 

8.46366 

.46799 
.47226 
.4p5o 
.48069 

4.68551 
.68551 
.68551 
.68551 
.68551 

4.68570 
.68570 
.68570 
.68570 
.68571 

5.31449 
.31449 
•3  '449 
.31449 
.31449 

5-3I430 
.31430 
.3M30 

'31429 

6300 
6360 
6420 
6480 
6540 

49 

'49304 
.49708 
.50108 

4.68551 
.68551 
.68550 
.68550 
.68550 

4.68571 
.68571 
.68572 
.68572 
.68572 

5-3I449 
•31449 

'3H50 

5-3I429 
.31429 
.31428 
.31428 
.31428 

6600 
6660 
6720 
6780 
6840 

50 
51 
52 
53 
54 

8.50504 

'51673 
.52055 

4.68550 
.68550 
.68550 
.68550 
.68550 

4.68572 
.68573 
.68573 
.68573 
.68573 

5.3I450 
.3H50 
.3H50 
.3H50 

5.31428 
.31427 
•31427 
.31427 
.31427 

6900 
6960 
7020 
7080 
7I4P 

11 
11 

59 

8.52434 
.52810 
.53183 
•53552 
.53919 

4.68549 
.68549 
.68549 
.68549 
.68549 

4.68574 
.68574 
.68574 
.68575 
•68575 

5.3I45I 
.3H5I 
.31451 
.3H5I 
.31451 

5.3I426 
.31426 
.31426 
.31425 

•31425 

7200 

60 

8.54282 

4.68549 

4-68575 

53H5I 

5  31425 

TABLE  IV.— LOGARITHMS,  ETC.  29 


TABLE    IV. 


LOGARITHMS 

OF  THE 

SINE,  COSINE,  TANGENT  AND  COTANGENT 

FOR 

EACH  MINUTE  OF  THE  QUADRANT. 


TABLE  IV. 


0° 

9 

L.  Sin. 

d. 

L.  Tang. 

c.  d. 

L.  Cotg. 

L.  Cos. 

Proj 

).  Pt 

S. 

0 

I 
2 

3 

4 

6.46373 
6  .  76  476 
6.94085 
7.06  579 

30x03 
17609 
12494 
9601 

6.46373 
o  .  76  476 
6.94085 
7.06579 

30103 
17609 
12494 
0601 

3.53627 
3.23524 
3  05915 
2.93421 

o.ooooo 
o.ooooo 
o.ooooo 
o.ooooo 

0.00000 

00 

3 

3 

.1 
.2 

•3 

3476 
348 
695 
1043 

3218 
322 
644 
965 

2997 

300 

599 
899 

i 
I 

9 

7.16  270 
7  24188 
7.30882 
7.36682 
7.41  797 

7918 
6694 
5800 

5»5 

4576 

7.16270 
7.24  188 
7.30882 
7.36682 
7.41  797 

7918 
6694 
5800 

S«5 

2.83730 
2.75812 
2.69  118 
2.63318 
2.58203 

o.ooooo 
o.ooooo 
o.ooooo 

0.00000 

o.ooooo 

55 
54 
53 
52 
5i 

•4 
•  5 

.x 

1390 
1738 

2&M 

280 

X287 

1609 

2633 
263 

1199 
1498 

2483 
248 

10 
ii 

12 
13 
14 

7.46373 
7.50512 
7-54291 
7-57  767 
7.60985 

4139 
3779 
3476 
32x8 

7-46373 
7.50512 
7.5429I 
7.57767 
7.60986 

4139 
3779 
3476 
3219 

2.53627 
2.49488 
2.45  709 
2.42233 
2.39014 

o.oo  ooo 
o.ooooo 
o.ooooo 
o.ooooo 
o.ooooo 

50 

42 
48 

47 
46 

.2 

•3 
•4 
•  5 

500 
84X 
ZI2X 
S401 

2227 

527 
790 
1053 

1316 

497 
745 
993 
1242 

1848 

ii 

Is7 
19 

7.63  982 
7.66  784 
7.69417 
7.71900 
7.74248 

2802 
2633 
2483 
2348 

7.63982 
7.66  ?8| 
7.69418 
7.71900 
7.74248 

2803 
2633 
2482 
2348 
2228 

2.36018 
2.33215 
2.30582 
2.28  100 
2.25  752 

0.00000 

o.ooooo 
9-99999 
9-99999 
9  99999 

45 
44 
43 
42 
41 

.x 

.2 

•3 
•  4 

.5 

223 
445 
668 
Sgx 
11x3 

202 

404 
606 
808 
XOIO 

370 
554 
739 
924 

20 

21 
22 
23 

24 

7.76475 
7-  78*>4 
7.80615 

7-82545 
7-84393 

2119 

2O2I 
X930 

x848 

7.76476 

7.78595 
7.80615 
7-82546 
7  84394 

2119 

2O2O 

1931 
1848 

2.23524 

2.21  405 
2.19385 
2.17454 
2.15  606 

9  99999 
9-99999 
9-99999 
9-99999 
9-99999 

40 

37 
36 

.1 

.2 

•3 

1704 

170 

5" 

1579 

158 

3x6 
474 

I47» 
«47 

294 
442 

11 
11 

29 

7.86  166 
7.87870 
7.89  509 
7  91088 
7.92  612 

1704 
x639 
»579 
«524 

7.86  167 
7-87871 
7.89510 
7.91  089 
7.92613 

1704 
1639 

»579 
1524 

2.13833 
2.12  129 
2  .  10  490 
2.08911 
2.07387 

9.99999 
9-99999 
9  99999 
9-99999 
9.99998 

35 
34 
33 

31 

•  4 
•5 

.1 

682 
852 

1379 
138 

632 

789 

1297 
130 

589 
736 

1223 

122 

30 

32 
33 
34 

7.94084 

7-95  5°8 
7.96  887 
7.98223 
7-99  520 

1472 
1424 
«379 
X336 
1297 

7.94086 
7-95  5io 
7.96889 
7-98225 
7.99522 

*473 
1424 
'379 
1336 
"97 

2.05914 
2.04490 
2.03  III 

2.01  77<j 
2.00478 

9.99998 
9.99998 
9.99998 
9.99998 
9.99998 

80 

S 

.2 

•3 
•4 
•5 

276 

552 

690 

11*8 

259 
389 
5*9 
649 

«45 

367 
489 

6x2 

39 

8.00  779 

8.02002 

8.03  192 
8.04350 
8.05478 

1259 

X223 
II9O 
XI58 
XI28 

8.00  781 
8.02004 
8.03  194 
8.04353 
8.05481 

1223 
1190 

"59 
1x28 

•99  219 
.97996 
.96806 
.95  647 
•945*9 

9.99998 
9.99998 
9-99997 
9-99  997 
9  99997 

25 
24 
23 

22 
21 

.x 

.2 

•3 
.4 
.5 

"5° 

116 
232 
347 
463 
570 

xxo 

220 

33° 
440 

55° 

105 
209 

418 

523 

40 

42 
43 
44 

8.06578 
8.07650 
8.08696 
8.09  718 
8.10  717 

IO72 
1046 
X022 

999 

n-rfi 

8.06581 
8.07653 
8.08  700 
8.09  722 
8  .  10  720 

1072 
1047 

XO22 

O?6 

•93  419 
.92347 
.91300 

9-99997 
9-99997 
9-99997 
9-99997 
9-99996 

20 

19 
18 

\l 

.1 

.2 

•  ^ 

999 

100 
200 
300 

954 
95 

191 

286 

914 
91 

183 
274 

46 
47 
48 

49 

8.11693 
8.12647 
8.13581 
8.14495 

8.I539I 

97° 

954 
934 
914 
896 

8.  ii  696 
8.12651 

8.13585 
8.14500 

8.15395 

97° 
955 
934 
9i5 
895 

0-0 

•  88304 
•87349 
.86415 
.85500 
.84605 

9.99996 
9.99996 
9.99996 

9-99996 

15 
13 

12 
II 

•4 
•  5 

.1 

400 
500 

877 

88 

382 
477 

843 

84 

366 
457 

812 
8x 

50 

5' 

52 
53 
54 

8.16268 
8.17  128 
8.17971 
8.18798 
8  19610 

°77 
860 

843 
827 
812 

8.16273 

8.I7I33 
8.17976 
8.18804 
8.19616 

860 
843 
828 
812 

•  83  727 
.82867 
.82  024 
.81  196 
.80384 

9-99995 
9  99995 
9-99995 
9  99995 
9  99995 

10 

I 

.2 

•3 
•4 
•5 

175 

263 

438 

169 
253 
337 
422 

162 
244 
325 
406 

3 

57 
58 
59 

8.20407 
8.21  189 
8.21  958 
8.22713 
8.23456 

797 
.    782 
769 
755 
743 

8.20413 
8.21  195 
8.21  964 

8.22  720 
8.23462 

797 
782 
769 

756 
742 

•79587 
.78805 
.78036 
.77280 
.76538 

9-99994 
9.99994 
9.99994 
9.99994 
9.99994 

5 
4 

2 
I 

.2 

•3 
•  4 

78 
«56 

755 

75 

226 

730 

73 
x46 
219 
292 

i  00 

8.24  186 

.73° 

8.24  192 

73° 

i  .  75  808 

9  99993 

0 

3  5 

L.  Cos. 

d. 

L.  Cotg. 

c.  d. 

L.  Tang. 

L.  Sin. 

9 

Pro] 

>.Pt 

S. 

89° 

LOGARITHMS  OF  SINE,  COSINE,  TANGENT  AND  COTANGENT,  ETC.     31 


1° 

9 

L.  Sin. 

d. 

L.  Tang. 

c.d. 

L.  Cotg. 

L.  Cos. 

Prop.  Pts. 

0 

I 

2 

3 
4 

8.24  186 
8.24903 
8.25609 
8.26304 
8.26988 

717 
7o6 
695 
684 
673 

663 
653 
644 
634 
624 

616 
608 
599 
590 
583 

575 
568 
560 
553 
547 
539 
533 
526 
520 
5H 
508 
502 
496 
491 
485 
480 
474 
470 
464 
459 
455 
450 
445 
441 
436 
433 
427 
424 
419 
416 
411 
408 
404 
400 
396 
393 
39° 
386 
382, 
379 
376 
373 
369 
367 
363 

8.24  192 
8.24910 
8.25616 
8.26312 
8.26996 

7o6 
696 
684 
673 
663 
654 
643 
634 
625 

617 
607 

599 

584 

575 
568 

553 
546 

54<> 
533 
527 
520 

509 
502 
496 

491 
486 

480 
475 
470 
464 
460 

455 
450 
446 

437 
432 
428 

424 

420 
416 
413 
408 

4<H 

401 

397 
393 

390 
386 
383 
380 

376 
373 
370 
367 
363 

.75808 
•  75090 

'73688 
•  73004 

9-99993 
9-99993 
9-99993 
9  99  993 
9-99992 

00 

59 
58 

P 

.X 

.2 

•3 

•4 
•  5 

.1 

.3 
•3 
•4 

•5 
.x 

.9 

•3 
•4 
•5 

.x 

.3 

•3 
•4 
•5 

.x 

.3 

•3 
•4 
•  5 

.1 

.9 

•3 
•4 
.5 

.1 

.3 

•3 
•4 
•5 

.1 

.3 

•3 
•4 

•5 

.2 

•3 

•4 
•  5 

717 

71.7 
143-4 
215.1 
286.8 
358-5 

653 

65-3 
130.6 

195.9 
261.9 
396.5 

599 

59-9 
119.8 
179.7 
939.6 
299.5 

553 

55-3 
110.6 
165.9 

221.2 

276.5 

5«-4 

102.8 

154.2 

205.6 

957.0 

480 

48 
96 

«44 
199 
940 

9» 

135 
1  80 
225 

420 

4* 

84 
126 
x68 

310 
390 

39 
78 
117 

'95 

695 

69-5 
139.0 
208.5 
278.3 
347  5 

634 

63-4 
126.8 
190.2 
253.6 
317.0 

583 

58.3 
116.6 

174-9 
933.2 
291.5 

539 

53-9 
107.8 
161.7 
215.6 
269.5 

Soa 

50.2 
100.4 
150.6 

200.8 

251.0 

470 

47 
94 

188 
*35 

^ 

i88 
132 
176 
220 

4IO 

41 
82 
123 
164 
205 

380 

38 
76 
1X4 

190 

673 

67.3 
134.6 
201.9 
269.3 
336.5 

6x6 

61.6 
123.  a 
184.8 
246.4 
308.0 

568 
56.8 
113.6 
170.4 
227.3 
284.0 

5a6 
Sa.6 
105.3 
X57-8 
310.4 
263.0 

490 
49 
98 
147 
196 

•45 

460 

46 
93 
«38 
184 
330 

430 

43 
86 

X29 

179 
915 

400 

40 

80 

120 
1  60 
20O 

370 

37 
74 

XXX 

148 
185 

9 
10 

ii 

12 

13 
14 

8.27661 
8.28324 

8.28977 
8.29621 
8-3025? 

8.27669 
8.28332 
8.28986 
8.29  629 
8  .  30  263 

•72331 
.71668 
.71014 
.70371 
•69  737 

9.99992 
9.99992 
9.99992 
9-99992 
9.99991 

55 
54 
53 
52 

8.30879 

8.31  495 
8.32  103 
8.32  702 
8.33292 

8.30888 
8-31  So? 

8.32  112 
8.327II 
8.33302 

.69  112 

9.99991 
9.99991 
9-99990 
9.99990 
9.99990 

50 

It 

19 

8-33  875 
8.34450 
8.35018 

8-35  578 
8.36131 

8.33886 
8.34461 
8.35029 

8-35  590 
8.36  143 

.66  114 

•65  539 
.64971 
.64419 
•63  857 

9.99990 
9-99989 

9.99989 
9.99989 

45 
44 
43 
42 
41 

20 

21 

22 

23 

24 

8.36678 
8.37217 
8.37750 
8.38276 
8.38796 

8.36689 
8.37229 
8.37  762 
8.38289 
8.38809 

•  633" 
.62  771 
.62  238 
.61  711 
.61  191 

9.99988 
9-99988 
9.99988 
9.99987 
9.99987 

40 

39 
38 

i 

29 

"30" 

32 
33 
34 

8.39310 
8.39818 
8.40320 
8.40816 
8.41  307 

8.39323 
8.39832 
8.40334 
8.40  830 
8.41  321 

.60677 
.60168 
.59666 
.59170 
•58679 

9.99987 
9.99986 
9.99986 
9.99986 

9.99984 
9.99984 

35 
34 
33 
32 

8.41  792 
8.42272 
8.42  746 
8.43216 
8.43680 

8.41  807 
8.42287 
8.42  762 
8.43232 
8.43696 

•58  193 
•57713 
•57238 
•  56  768 
•  56304 

30 

29 
28 

11 

35 

36 

39 

8.44  139 
8-44594 
8.45044 
8.45489 
8-45  93° 

8.44  156 
8.44611 
8.45061 

8-45  507 
8.45948 

-55844 
•55389 
•54939 
•54493 
•54052 

9-99983 

9.99982 
9.99982 

25 
24 

23 

22 
21 

40 

42 
43 
44 

8.46366 
8.46  799 
8.47  226 
8.47650 
8.48069 

8.46385 
8.46817 

8.47245 
8.47669 
8.48089 

.53615 
•53  183 
•52  755 
•52331 
•  Si  9ii 

9.99982 
9-9998I 
9.99981 
9.99981 
9.99980 

20 

18 

\l 

1 

49 

8.48485 
8.48896 
8.49304 
8.49  708 
8.50  108 

8.48505 
8.48917 
8.49325 
8.49  729 
8.50  130 

•51  495 
•  5i  083 
•50675 
.50271 

•49  870 

9.99980 
9-99979 
9-99979 
9-99979 
9.99978 

15 
14 
13 

12 
II 

50 

1  5I 
52 

53 
59 

8.50  504 
8.50  897 
8.51287 

8.51  673 
8.52055 

8-50527 
8.50920 
8.51  310 
8.51  696 
8.52079 

•49473 
.49  080 
.48  690 
.48  304 
•47  921 

9.99978 
9-99977 
9-99977 
9-99977 
9.99976 

10 

§ 
i 

5 
4 
3 

2 

I 

8.52434 
8.52810 
8-53  183 
8.53552 
8.53919 

8.52459 
8.52835 
8.53  208 
8-53578 
8.53945 

•47  541 
•47  165 
.46  792 
.46422 
•46055 

9.99976 
9-99975 
9-99975 
9-99974 
9-99974 

00 

8.54282 

8.54308 

1.45692 

9-99974 

0 

L.  Cos. 

(1. 

L.  Cotg. 

c.d. 

L.  Tang. 

L.  Sin. 

/ 

Prop.  Pts. 

88° 

TABLE  IV. 


2° 

t 

L.  Sin. 

d. 

L.  Tang. 

c.d. 

L.  Cotg. 

L.  Cos. 

Pro] 

).  PtS 

>. 

0 

I 

2 

3 
4 

8.54282 
8  .  54  642 
8-54999 
8-55354 
8-55  705 

360 

357 
355 
35i 
349 

8.54308 
8.54669 
8.55027 
8.55382 
8-55734 

361 
358 

355 
352 
349 

1.45692 
I.4533I 
•44973 
.44618 
.44266 

9-99974 
9-99973 
9-99973 
9.99972 
9.99972 

GO 

59 

58 

11 

.1 

.2 

•3 

360 
36 

72 

108 

350 

35 
70 
105 

340 

3* 

a 

102 

I 

9 

8.56054 
8  .  56  400 

8.56743 
8.57084 
8.57421 

346 
343 
341 
337 
336 

8.56083 
8.56429 

8.56773 
8.57114 

8-57452 

346 
344 
34i 
338 

006 

.43917 
•43  571 
•  43  227 
.42886 
•  42  548 

9.99971 
9.99971 
9.99970 
9.99970 
9.99969 

55 
54 
53 
52 
5i 

•4 
•  5 
.6 

•7 
.8 

144 

180 
216 
252 
288 

140 

»75 

210 

245 
280 

I36 
170 
204 
238 
272 

10 

it 

12 
13 

14 

8-57757 
8  .  58  089 

8.58419 
8.58747 
8.59072 

333 
330 
328 
325 
323 

8.57  788 
8.58121 

8  58451 
8.58779 
8.59105 

333 
330 
3?8 
y€ 

32* 

.42  212 
.41  879 

•41  549 

.41  221 

.40895 

9.99969 
9.99968 
9-99968 
9.99967 
9-99967 

50 

8 
% 

•9 
.x 

.3 

•3 

324 
330 

33 
66 

99 

315 
320 
32 
64 
96 

306 
310 
3* 

6a 
93 

15 
10 
17 

18 
19 

8-59395 
8.59715 
8.60033 
8.60349 
8.60662 

320 
3i3 
316 
313 
311 

8.59428 

8  59749 
8.60068 
8  .  60  384 
8.60698 

321 
319 
316 
314 

.40572 
.40251 
-39932 
.396l6 
.39302 

9-99967 
9.99966 
9.99966 

9-99965 
9.99964 

45 
44 
43 
42 
41 

*4 
•5 
.6 

•  7 

.8 

132 
165 
108 
231 
264 

128 

160 
192 
224 
256 

124  1 

»55 

186  . 
217 
248  . 

20 

21 
22 

23 
24 

8.60973 
8.61  282 
8.61  589 
8.61  894 
8.62  196 

309 
307 
305 
302 

8.61  009 
8.61  319 
8.61  626 
8.61  931 
8.62234 

310 
307 
305 
303 

.38991 
.38681 

.3f374 
.38069 
•37  766 

9.99964 
9.99963 
9.99963 
9.99962 
9.99962 

40 

39 
38 

y 

•y 

.X 

.a 

•3 

297 
300 
30 
60 
90 

288 
290 

29 
58 
87 

27$ 
285 
28.- 
57-< 
85-5 

s 

3 

29 

8.62497 
8.62  79! 
8.63091 
8.63385 
8.63678 

298 
296 
294 
293 

8-62  535 
8.62834 
8.63  131 
8.63426 
8.63  718 

299 
297 

295 
292 

.37465 

.37  166 
36869 

.36574 
.36282 

9.99961 
9.99961 
9.99960 
9.99960 
9-99959 

35 
34 
33 
32 
3* 

•4 
.5 
.6 

•  7 

.8 

120 
ISO 

180 

210 

240 

116 
145 
174 
203 
232 

114.0 
142.5 
171.0 

199.5 
228.0 

BO 

3i 
32 
33 
34 

8.63968 
8.64256 
8.64543 
8.64827 
8.65  no 

288 
287 
284 
283 
281 

8.64009 
8.64298 
8.64585 
8.64870 
8.65  154 

289 
,.87 
"85 
284 
281 

•35  99i 
•35  702 
•35415 
•35  130 
.34846 

9-99959 
9.99958 
9-99958 
9-99957 
9.99956 

80 

3 

11 

•9 
.x 

.2 

•3 

270 
280 

28.0 
S6.0 
84.0 

275 

27.5 
55-0 
82.5 

256.5 
270 
27.0 
54-0 

81.0 

35 
36 

* 

1  39 

8-65391 
8.65670 
8.65947 
8.66223 
8.66497 

279 
277 
276 
274 

8.65435 
8.65  715 
8.65993 
8.66269 

8.66543 

280 
278 
276 
274 

.34565 
•34285 
.34007 
•33  73i 
•33457 

9.99956 
9-99955 
9-99955 
9-99954 
9-99954 

25 
24 

23 

22 
21 

•4 
•  5 
.6 

•7 
.8 

112.  0 

I40.O 

168.0 
196.0 
224.0 

IIO.O 

137.5 

165.0 

192.5 

22O.O 

108.0 
135-0 
162.0 
189  o 

2ld   0 

40 

4i 

42 

43 
44 

8.66769 
8.67039 
8.67308 

8.67575 
8.67841 

270 
269 
267 
266 
267 

8.66816 
8.67087 
8.67356 
8.67624 
8.67890 

271 
269 
268 
266 
_g. 

•33  184 
•32913 
.32644 
.32376 
.32  no 

9-99953 
9.99952 
9-99952 
9-99951 
9-99951 

20 

JQ 

!2 

.1 

.2 

•3 

265 

.26.5 
•53-0 
•79-5 

260 

.26.O 
.52.0 
.78.0 

255 

•25.5 
.51.0 

.76.5 

9 

% 

49 

8.68  104 
8.68367 
8.68627 
8.68886 
8.69  144 

263 
260 

259 

258 

2-5 

8.68  154 
8.68417 
8.68678 
8.68938 
8.69  196 

263 
261 
260 
258 

.31  846 

•31  583 
.31  322 
.31  062 
•  30804 

9-99950 
9-99949 
9.99949 
9.99948 
9.99948 

15 

H 
13 

12 
II 

-4 

•  5 
.6 

•7 
.8 

132-5 
159.0 
185-5 

212.0 

2^8  <; 

I04.O 
130.0 
156.0 
182.0 
208.0 

127.5 
153.0 
178.5 

204.0 

50 

5i 

52 
S3 

54 

8.69400 
8.69654 
8.69907 

8.70159 
8.70409 

254 
253 
253 
250 

8.69453 
8.69708 
8.69962 
8.70214 
8.70465 

25S 
254 
252 
251 

.30547 
.30292 
.30038 
.29786 
•29535 

9  99947 
9.99946 
9.99946 
9-99945 
9-99944 

10 

1 
I 

.1 
.a 
•3 

250 
.25.0 
.50.0 
.75-0 

345 
.24.5 
.49.0 
•73-5 

240 
.24.0 

.48.0 
.72.0 

56 

Ji 

59 

8.70658 
8.70905 
8.71  151 
8.71395 
8.71  638 

949 

247 
246 
344 
243 

8.70714 
8.70962 
8.71  208 

§•71453 
8.71697 

249 
248 
246 
245 
244 

.29286 
.29038 
.28  792 
•  28  547 
.28  303 

9-99944 
9-99943 
9.99942 
9.99942 
9.9994: 

5 
4 
3 

i 

.4 

•5 
.6 

•7 

125.0 
150.0 
175.0 

200.0 
295.O 

122.5 
X47.0 
X7I-5 
Z96.O 
22O.5 

I2O.O 

144.0 
168.0 

193.0 
2x6.0 

GO 

8.71  880 

8.71940 

1.28060 

9.99940 

0 

L.  Cos. 

d. 

L.  Cotg. 

c.d. 

L.  Tang. 

L.  Sin. 

t 

Pro 

P.    Pfc 

i. 

1 

87° 

LOGARITHMS  OF  SINE,  COSINE,  TANGENT  AND  COTANGENT,  ETC.     33 


3° 

t 

L.  Sin. 

d. 

L.  Tang. 

e.d. 

L.  Cotg. 

L.  Cos. 

P£0p.  PtS. 

0 

I 

8.71880 

8.72  120 

340 

8.71940 
8.72  181 

241 

1.28060 
1.27819 

9.99940 
9.99940 

GO 

338 

334 

339 

2 

8.72359 

339 

2-0 

8  .  72  420 

239 

1.27580 

9-99939 

58 

.1 

33.8 

83-4 

33.9 

3 
4 

8.72$97 
8.72834 

330 

837 

8.72659 
8.72896 

239 
237 

2T.6 

1.27341 
1.27  104 

9-99938 
9-99938 

y 

.3 

•  3 

47.6 

46.8 
70.2 

45-8 
68.7 

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8.73069 
8.73303 

234 

8.73132 
8.73366 

1.26868 
1.26634 

9  99937 
9.99936 

55 
54 

-4 

•  5 

95-2 
119.0 

93-6 
117.0 

91.6 
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I 

8-73535 
8.73767 

232 
232 

8.73600 
8.738J2 

234 

232 

i  .  26  400 
1.26  1  68 

9-999J6 
9-99935 

53 
52 

.6 
•7 

142.8 
166.6 

140.4 
163.8 

137-4 
160.3 

if 

8-73997 

229 
228 

8.74063 

231 

229 

1-25937 

9  99934 

51 

.8 
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190.4 
314.2 

187.2 

2X0.6 

183.3 
206.  x 

8.74226 

8.74292 

1.25  708 

9  99934 

50 

ii 

12 

8-74454 
8.74680 

226 

226 

8.74521 
8.74748 

229 

227 

__£ 

1-25479 
1.25  252 

9  99933 
9-99932 

4-Q 

., 

22.5 

22O 
32.0 

310 
3X.6 

13 

8.74906 

8.74974 

1.25  026 

9.99932 

47 

.3 

45-0 

44.0 

43-a 

H 

8.75130 

224 

8-75  199 

225 

i  .  24  801 

9-99931 

46 

•3 

67-5 

66.0 

64.8 

16 

8-75353 
8-75575 

222 

8.75423 
8.75645 

222 

1-24577 
1-24355 

9.99930 
9  99  9-9 

45 

44 

•4 

•  5 

90.0 
112.5 

88.0 

XIO.O 

86.4 
108.0 

ii 

19 

8-75795 
8.76015 
8.76234 

220 
8X9 
217 

8.75867 
8.76087 
8.76306 

220 
2X9 

1.24133 
1.23913 

1.23694 

9  99929 
9.99928 
9.99927 

43 
42 

41 

.6 

•  7 
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135-0 
157.5 
180.0 

133.0 

154.0 

176.0 

_-0    n 

139.6 
151.2 
173.8 

20 

21 

22 

8.76451 
8.  76667 
8.76883 

216 

3l6 

8.76525 
8.76742 
8.76958 

3J7 

2x6 

1-23475 
1.23258 
1.23  042 

9-99926 
9.99926 
9.99925 

40 

39 
38 

•9 
.1 

313 

21.2 

198.0 

308 

20.8 

304 

20.4 

23 

8.77097 

8X4 

8.77173 

8X5 

1.22  827 

9.99924 

37 

.3 

42.4 

41.6 

40.8 

24 

8.77310 

213 

8.77387 

8X4 

I.226I3 

9.99923 

36 

•3 

63.6 

63.4 

61.3 

25 
26 

8.77522 
8-77  733 

311 

8.77600 
8.77811 

813 
211 

I  .  22  400 
1.22  189 

9.99923 
9.99922 

35 

S4 

•4 
•5 

84.8 

1  06.0 

83.2 

104.0 

81.6 

103.0 

% 

29 

8-77943 
8.78152 
8.78360 

309 

208 
208 

8  .  78  022 
8.78232 
8.78441 

211 
2IO 
209 
208 

I.  21  978 
1.  21  768 

I-2I559 

9.99921 
9  99  920 
9.99920 

33 
32 

w 

.6 

•7 

.8 

•9 

127.2 
148.4 
169.6 
190.8 

124.8 

145.6 
166.4 

187.2 

122.4 
142.8 
163.2 
183.6 

30 

8.78508 

8.78649 

I-  21  351 

9.99919 

S2 

8.78774 
8.78979 

205 

8.78855 
8.79061 

206 
206 

I   21  145 
1.20939 

9.99918 
9.99917 

28 

.1 

301 

2O.  I 

197 
19.7 

193 

19.3 

33 

34 

8.79183 
8.79386 

204 

203 

2O2 

8  .  79  266 
8.79470 

205 
204 

1.20734 
1  .  20  530 

9  99917 
9.99916 

z 

.3 

•3 

40.2 
60.3 

39-4 
59-1 

38.6 
57-9 

35 
36 

9 

39 
10" 

8.79588 
8.79789 
8.79990 
8.80  189 
8.80388 

201 
201 
199 
199 
197 
197 

8.79673 

8.79875 
8  80076 
8.80277 
8.80476 

203 

202 
201 
201 
199 
I98 

x98 

1.20327 
1.20  125 
1  .  19  924 
1  .  19  723 
1  .  19  524 

9-999I5 
9.99914 

9  999'3 
9-999I3 
9.99912 

25 
24 
23 

22 
21 

20" 

19 

•4 
•5 
.6 

•  7 

.8 

•9 

100.5 

120.6 

140.7 
160.8 
180.9 

189 

98.5 
1x8.2 

137-9 

157.6 
185 

77«2 
96-5 
1x5.8 
I35.I 
154-4 
173-7 
181 

8.80585 
8.80782 

8  80674 
8.80872 

1  .  19  326 
I.I9  128 

9.99911 
9.99910 

42 

8.80978 

196 

8.81068 

196 

1.18932 

9.99909 

18 

•  i 

18.9 

18.5 

18.1 

43 

8.81  173 

X95 

8.81  264 

196 

I.I8736 

9.99909 

17 

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37.  « 

37.0 

36.3 

44 

8.81  367 

194 

8.81459 

195 

1.18  541 

9.99  908 

16 

•3 

56.7 

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55-5 

54-3 

45 
46 

49 

8.81  560 
8.81  752 
8.81  944 
8.82  134 
8.82324 

192 
192 
190 
190 

1  80 

8.81  653 
8  81  846 
8.82038 
8.82  230 
8.82420 

194 

193 
192 
192 
190 

1.18347 
1.18154 
1.17962 
1.17  770 
1.17  580 

9.99907 
9.99906 

9-99905 
9.99904 
9.99904 

15 
13 

12 
II 

•4 
•  5 
.6 

•  7 

.8 
•  0 

75.0 
94-5 

132.3 
151.9 
170.1 

74.0 
92.5 

XXX.  0 

129.5 
148.0 
166.5 

73.4 
90.5 
108.6 
126.7 
144-8 
162.0 

50 

8.82513 
8.82  701 

188 

8.82610 
8.82  799 

190 
x89 

1.17390 

I.I7  201 

9.99903 
9.99  902 

10 

4 

3       a       x 

52 
53 

8.82888 
8.83075 
8.83261 

187 
x87 

186 

8.82  987 
8.83  175 
8.83361 

1  88 

188 
1  86 

-QJC 

I.I70I3 

i  .  16  825 
i  .  16  639 

9.99901 
9.99  900 
9.99899 

I 

.1 

.3 

•3 

0.4 
0.8 

.2 

0.3      O.2      O.X 

0.6    0.4    o.a 
0.9    0.6    0.3 

59 

8.83446 
8.83630 
8.83813 
8.83996 
8.84177 

184 
183 
183 
181 
x8x 

8-83547 
8-83732 
8.83916 
8  .  84  100 

8.84282 

185 
184 
x84 
182 
182 

i  •  16  453 
1  .  16  268 
i  .  16  084 
1.15900 
1.15  718 

9-99898 
9.99898 
9.99897 
9.99896 
9-99895 

5 
4 
3 

2 
I 

•4 

•5 
.6 

•  7 
.8 
•9 

.O 

•  4 

.8 

.2 

3.6 

1.5      .0    0.5 
x.8      .2    0.6 
a.  i      .4    0.7 
3.4      .608 
9.7      .809 

60 

8.84358 

8  .  84  464 

I-I5536 

9.99894 

o 

L.  Cos. 

d. 

L.  Cotg. 

c.d 

L.  Tang. 

L.  Sin. 

f 

Prop.  Pts. 

86° 

TABLE  IV. 


4° 

t 

L.  Sin. 

d. 

L.  Tang. 

c.d. 

L.  Cotg. 

L.  Cos. 

Prop.  Pts. 

0 

I 

2 

3 
4 

8.84358 
8.84539 
8.84  718 
8.84897 
8.85075 

181 
179 
179 
178 
177 
177 
176 
US 
175 
173 
173 
173 
171 
171 
171 
169 
169 
169 
167 
168 

166 
1  66 
165 
164 
164 

163 
163 
162 
162 
160 
161 
»59 
J59 
159 
158 

157 
157 
156 
155 
155 
»55 
154 
153 
*52 
152 
152 
151 
15* 
150 
150 
149 
149 
148 
147 
M7 

147 
146 
146 
»4S 
145 

8.84464 
8.84646 
8.84826 
8.85006 
8.85  185 

182 
180 

180 
179 
178 

•15536 
•15354 
•15  *74 
.14994 
.14815 

9.99894 

9-99893 
9.09892 
9.99891 
9.99891 

60 

11 
H 

.x 

.2 

•3 
.4 
-5 
.6 
•  7 
.8 

•9 
.1 

.2 

•3 
•  4 
•5 
.6 

•  7 
.8 

•9 
.t 

.2 

•3 
•  4 
•5 
.6 

•7 
.8 

•9 

") 
.i 

.2 

•3 
•  4 
•  5 
.6 

•7 
.8 

•9 
.1 

.2 

•3 

•  4 
•  5 
.6 

•  7 
.8 

•9 

.2 

•  3 
•4 
•5 
.6 

•  7 
.8 

•9 

z8z 

18.1 
36.2 

54-3 
72.4 
90.5 
108.6 
126.7 
144-8 
162.9 

175 
i7-5 
35-0 
53-5 
70.0 
87-5 
105.0 
122.5 
140.0 
157-5 
168 
.16.8 
33-6 
50.4 

779 

17.9 
35-8 
53-7 
71.6 

89-5 
107.4 

125-3 
143.2 

161.1 

173 

17-3 
34-6 
51-9 
69.2 
86.5 
103.8 

121.  1 
138.4 
155-7 

166 
16.6 

/33-2 
49.8 
66.4 
83.0 
99.0 
116.2 

132.8 
149.4 

159 
15.9 

31-8 
47-7 
63.6 
79-5 
95-4 
111.3 
127.2 
i43-» 
153 
15-3 
30.6 

45-9 
61.2 

76.5 
91.8 
107.1 
122.4 
"37-7 
M7 
14.7 
29.4 
44.1 
58.8 

73-5 
88.2 
102.9 
117.6 
132.3 

177 

17.7 
35-4 
53-1 
70.8 
88.5 
106.9 
123.9 
141.6 
*59-3 
171 
17.1 
34-» 
Si-3 
68.4 
85.5 

102.6 

119.7 

136.8 

153-9 

164 
16.4 

32.8 
49.8 
65.6 
820 
98-4 
114.8 
131-3 
147.6 

157 
15-7 
31-4 

47-» 
62.8 

78-5 
94-3 
109.9 
125.6 
MI-  3  ' 
IS* 
15.1 
30.2 

45-3 
60.4 
75-5 
90.6 
105.7 

120.8 

135.9 

z 

O.I 
O.S 

0.3 
0.4 
0.5 

0.6 
0.7 
0.8 
0.9 

I 
I 

9 

8.85  252 
8.85429 
8.85  605 
8.85  780 
8.8595! 

8-85  363 
8.85  540 
8.85  717 
8.85893 
8.86069 

177 
177 
176 
176 
174 
174 
174 
172 
172 
171 
171 
170 
169 
169 
168 

167 
167 
1  66 
165 
165 

165 
163 
163 

163 
161 
162 
160 
1  60 
1  60 
!59 
158 
158 
157 
157 
156 
155 
155 
155 
153 
154 

152 
152 
152 
151 
151 
150 
'So 
149 
148 
149 

147 
H7 
147 
146 
146 

•  H  637 
.14460 
.  14  283 
.14107 
•I3931 

9.99890 
9.99  889 
9.99888 
9-99887 
9.99886 

55 
54 
53 
S2 
5i 

10 

ii 

12 
13 

14 

8.86  128 
8.86301 

8.86474 
8.86645 
8-86816 

8.86243 
8.86417 
8.86591 
8.86  763 
8.86935 

•13757 
•  !3  583 
•  13  409 
.13237 
.13065 

9.99885 
9.99884 
9.99883 
9.99882 
9.99881 

50 

49 
48 
47 
46 

15 

1  6 

\l 

I  I9 

8.86987 
8.87  156 
8.87325 
8.87494 
8.87661 

8.87  106 
8.87277 

8.87447 
8.87616 

8.87785 

.12894 

.  12  723 

.12553 
.  12  384 
.12  215 

9.99880 
9.99879 
9.99879 
9.99878 
9.09877 

45 

£4. 

43 
42 

4i 

20 

21 
22 

23 
24 

8.87829 

8.87995 
8.88  ?6i 
8.88326 
8.88490 

8-87953 
8.88  126 
8.88287 
8.88453 
8.88618 

.  12  047 
.11  880 
.11  713 

•II547 
.11  382 

9-99876 

9  99875 
9.99874 

9.99873 
9.99872 

40 

i§ 
H 

2 

II 

£ 

31 
32 

33 
34 

8.88654 
8.88817 
8.88980 
8.89  142 
8.89304 

8.88  783 
8.88948 
8.89  in 
8.89274 
8.89437 

.11  217 
.  1  1  052 

.  10  889 
.  10  726 
.  10  563 

9.99871 
9.99870 
9  .  99  869 
9.99868 
9.99867 

35 
34 
33 
32 
3i 

67.5 
84.0 
100.8 
117.6 
134-4 

8.89464 
8.89625 
8.89  784 
8.89943 

8.90  102 

8.89598 
8.89  760 
8.89920 
8.90080 
8.90240 

.  10  402 
.  10  240 
.  10  080 
.09920 

.09760 

9.99866 
9.99865 
9.99864 
9.99863 
9  .  99  862 

30 

29 
28 

% 

151.2 

;  162 

16.2 

33-4 
48.6 
64.8 

81.0 
97.2 
"3-4 
129.6 
145-8 
155 
iS-5 
31-0 
46.5 
62.0 

77-5 
93-o 
108.5 
124.0 
139-5 
149 
14.9 
29.8 
44-7 
59-6 
74-5 
89.4 
104.3 
119.2 
I34-J 

it 
3 

39 

8.9O260 
8.90417 

8.90  574 
8.90  730 
8.90885 

8.90399 

8.90557 
8.90  715 
8.90872 
8.91  029 

.09601 

.09443 

.09285 
.09  128 

.08  971 

9.99861 
9.99860 

9-99859 
9.99858 

.9  99857 

25 

24 

23 

22 
21 

"20" 

ii 

'to 

i  4i 

i  4* 
43 
44 

8.91  040 
8.91  195 
8.91  349 
8.91  502 
8.91  655 

8.91  185 
8.91  340 
8.91  495 
8.91  650 
8.91  803 

.08815 
.08  660 
.08  505 
.08350 
.08  197 

9.99856 
9.99855 
9-99854 
9.99853 
9.99852 

9 
9 

49 

8.91  807 
8.91  959 
8.92  no 
8.92261 
8.92411 

8-91  957 
8.92  no 
8  .  92  262 
8  92  414 
8.92565 
8.92  716 
.8.92866 
8.93016 
8.9.3  165 
8-93  313 

.08043 
.07890 

.07  738 
.07  586 
07435 

9.99851 
9.99850 
9-99848 
9.99847 
9.99846 

15 

14 
13 

12 
II 

50 

Si 

52 
53 

54 

8.92  561 
8.92  710 
8.92859 
8.93007 
8-93  154 

.07  284 
•07  134 

.06  984 

.06  835 
.06  687 

9-99845 
9.99844 
9-99843 
9.99842 
9.99841 

10 

1 
I 

55 
56 

R 

59 

8.93301 
8.93448 

8-93594 
8.93  740 
8.93  885 

8.93462 
8  9^609 
8-9.3756 
8.93903 
8.94049 

.06  538 
.06  391 

.06244 

.06  097 
05951 

9.99840 

9-99839 
9.99838 

9.99837 
9.99836 

5 
4 
3 

2 

I 

60 

8.94030 

8.94195 

1.05805 

9-99834 

0 

L.  Cos. 

(1. 

L.  Cotg. 

1C.  d. 

L.  Tang. 

L.  Sin. 

f 

Prop.  Pts. 

85°                          .             | 

LOGARITHMS  OF  SINE,  COSINE,  TANGENT  AND  COTANGENT,  ETC.     35 


5° 

9 

L.  Sin. 

d. 

L.  Tang-. 

c.  d. 

L.  Cots. 

L.  Cos. 

Pro 

p.  Pfe 

1. 

0 

I 

2 

3 
4 

8.94030 
8-94  174 
8.94317 
8.94461 
8.94603 

144 
MS 

144 
142 

8-94  195 
8.94340 

8.94485 
8.94630 
8-94  773 

145 
145 
145 

«43 
144 

•05805 
.05  660 
•05  5i5 
•05  370 
.05  227 

9-99834 
9-99833 
9.99832 
9.99831 
9.99830 

60 

3 
3 

.X 

.3 

•3 

145 

14.5 

ay  o 
43-5 

143 

«4-3 
38.6 
43.9 

14X 
X4.x 

28.9 
42-3 

\ 

I 

9 

8.94746 
8.94887 
8.95  029 
8.95  170 
8-95  3io 

»4» 
142 

«4« 
140 

8.94917 
8.95060 

8.95  202 

8.95  344 
8.95  486 

143 
142 
142 
143 

.05  083 
.04  940 
.04  798 
.04  656 
•04  5H 

9.99829 
9.99828 
9-99827 
9.99825 
9-99824 

55 
54 
53 
52 
5i 

•4 
•5 

.6 

•7 
.8 

58.0 
73.5 
87.0 

101.5 
116.0 

57-2 
71.5 
85.8 

1  00.  1 

1x4.4 

56.4 
70.5 
84.6 
98.7 

XI3.8 

10 

ii 

12 
13 
14 

8-95  450 
8-95  589 
8.95  728 
8.95  867 
8.96005 

139 
139 
139 
138 

178 

8.95  627 
8.95  767 
8.95908 
8.96047 
8.96  187 

140 
141 

139 
140 

x?8 

•04  373 
•04  233 
.04092 

•03  953 
.03  813 

9.99823 
9.99822 
9.99821 
9.99820 
9.99819 

50 

2 
3 

•9 

.X 
.9 

•3 

I30-5 
139 

13-9 
37.8 

41-7 

128.7 

138 
13.8 
«?.« 

4«-4 

126.9 

136 
13.6 

07.9 

40.8 

!i 

ii 

19 

8.96143 
8.96280 

8.96417 
8.96  553 
8.96689 

»37 
»37 
136 
136 

,,6 

8-96325 
8.96464 
8.96602 
8.96  739 
8.96877 

139 
138 
«37 
138 

TQ6 

•03  675 
•03  536 
•03  398 
.03  261 
.03  123 

9-99817 
9.99816 
9.99815 

9-99814 
9.99813 

45 
44 
43 
42 

41 

•4 
•5 

.6 

•7 
.8 

55-6 
69.S 
83-4 
97-3 

XII.  3 

55-2 
69.0 
82.8 
96.6 
1x0.4 

54-4 
68.0 

81.6 
95.  a 
1  08.  8 

20 

21 
22 
23 
24 

8.96825 
8.96960 
8.97095 
8.97229 
8.97363 

«35 
135 
134 
134 

8.97013 
8.97  ijo 
8.97285 
8.97421 
8.97556 

»37 

135 
136 
J35 

.02  987 
.02  8^0 
.02  7lS 

.02  579 
.02444 

9.99812 
9.99  810 
9.99809 
9.99808 
9.99807 

40 

P 
II 

•9 
.x 

.9 

•3 

125.1 

135 

«3-5 
37.0 
40.5 

124.2 
133 

»3-3 
26.6 

39-9 

122.4 
131 
X3.x 
26.9 
39-3 

3 

3 

29 

8.97496 
8.97629 
8.97  762 
8  .  97  894 

133 
133 
132 
132 

8.97691 
8.97825 

8-97959 
8.98092 
8.98225 

134 
134 
133 
133 

.02309 

•02  175 
.02  041 

.01  908 

•oi  775 

9.99806 
9-99804 
9.99803 
9.99802 
9.99801 

35 
34 
33 
32 
3i 

•4 
•5 
.6 

•  7 
.8 

S4-o 
67.5 
81.0 
94-5 

108.0 

53-2 
66.5 
79-8 
93-i 
ro6.4 

Sa-4 
6S.S 
78.6 
9X.7 
104.8 

ao 

31 
32 

33 

34 

8.98157 
8.98288 
8.98419 
8.98  549 
8.98679 

131 
131 
130 
130 

8.98358 
8.98490 
8.98622 
8.98  753 
8.98884 

132 
132 
131 
131 

.01  642 
.01  510 
.01  378 
.01  247 
.01  116 

9.99800 
9  99  798 
9-99  797 
9.99796 

9-99795 

30 

% 

Vw 
.9 

•  3 

129 
13.9 
25.8 
38.7 

.19.7 
128 

X3.8 

25.6 
38.4 

117.9 

126 

X3.6 

25.9 

37-8 

1 

39 

8.98808 

8.98937 
8.99066 

8-99  194 
8.99322 

129 
129 
128 
128 
128 

8.99015 
8.99145 
8.99275 
8.99405 
8-99534 

130 
130 
130 
129 
128 

.00985 
.00855 
.00  725 

•00595 

.00466 

9-99793 
9  99  792 
9-99  79i 
9.99  790 
9.99  788 

25 
24 
23 

22 
21 

•4 
•5 
.6 

•  7 

.8 

51.6 
64.5 
77-4 
90.3 
103.2 
116  x 

51.2 
64.0 
76.8 
89.6 

:03.4 

50.4 
63.0 
75-6 
88.3 
100.8 

40 

4i 
42 

43 
44 

8.99450 
8-99577 
8-99  704 
8.99830 
8.99956 

127 
127 
126 
126 
126 

8.99662 
8.99  791 
8.99919 
9.00046 
9.00  174 

129 
128 
127 

128 

.00338 

.00209 
.00081 

0.99954 
0.99826 

9-99  787 
9.99  786 

9.99785 
9-99  783 
9.99  782 

20 

!l 

.x 

.3 
•3 

MS 

12.5 
35.0 
37-5 

123 

13.3 
34.6 
36.9 

123 
12.2 
24-4 
36.6 

.0     0 

3 

4^ 

1  49 

9.00082 
9.00207 
9.00332 
9.00456 
9.00  581 

125 
125 
124 
125 

9.00301 
9.00427 

9-00553 
9.00679 
9.00  805 

126 
126 
126 
126 

0.99699 

0-99  573 
0.99447 
0.99321 
0.99  195 

9.99  781 
9-99  78o 
9-99  778 
9-99  777 
9-99  776 

15 
H 
13 

12 
II 

•4 
•  5 
.6 
•7 
.8 

50.0 
62.5 
75-o 
87.5 

100.  0 
"13-5 

49.2 

61.5 

73-8 
86.1 
98.4 

*IO.7 

61.0 
73-  « 
85-4 
97.6 

IOQ.8 

50 

Si 

52 
53 
54 

9.00  704 
9.00828 
9.00951 
9.01  374 
9.01  196 

124 
123 
123 

122 

9.00  930 
9-01  055 
9.01  179 
9-Oi  303 
9.01  427 

125 
124 
124 

124 

0.99070 
0.98  945 
0.98821 
0.98697 
0.98573 

9-99  775 
9-99773 
9-99  772 
9-99  77i 
9-99  769 

10 

1 
I 

.x 

.3 

•3 

121 
13.  1 
24.2 
36.3 

.0      . 

120 
12.0 

*4.o 

36.0 

.0    _ 

X 
O.I 
O.3 
0.3 

11 
11 

59 

9.01  318 
9.01  440 
9.01  561 
9.01  682 
9.01  803 

122 
121 
I2X 
121 

9-Oi  550 
9.01  673 
9.01  796 
9.01  918 
9  .  02  040 

123 

123 

122 
122 

0.98  450 
0.98  327 
0.98  204 
o  98  082 
0.97  960 

9.99768 
9.99  767 
9  99  765 
9-99764 
9.99763 

5 
4 
3 

2 
I 

•4 
•  5 
.6 

•7 
.8 
•  0 

48.4 
60.5 

72.6 
84.7 
96.8 
108.9 

60.0 
72.0 

84.a 
96.0 
108.0 

0-5 

0.6 
f      0.7 
08 
o.o 

GO 

9.01  923 

9.02  162 

0.97838 

9-99  76i 

0 

L.  Cos. 

d. 

L.  Cotgf. 

c.d. 

L.  Tang. 

L.  Sin. 

f 

Pro 

P.  Pfc 

L 

84° 

6                                                                               TABLJ2,  IV. 

6° 

I 

L.  Sin. 

(1. 

L.  Tang. 

c.d. 

L.Cotg. 

L.  Cos. 

Prop.  Pts. 

0 

I 

2 

3 
4 

I 
I 

9 

lo- 
ii 

12 

13 

14 

9-oi  923 
9.02  043 
9.02  163 
9.02283 
9.02402 

120 
1  2O 
120 
119 

118 
119 
118 
117 
118 
117 
117 
116 
116 
"6 
116 

"5 
"5 
114 
"5 
"3 
114 
114 
"3 

112 
XI3 
112 
112 
ZI2 
XII 
III 
III 
110 
1  10 
110 
110 
109 
109 
109 

108 
109 
108 
107 
108 
107 
107 
106 
107 
106 
105 
106 
105 
105 
105 
105 
104 

104 
104 
103 
103 
103 

9.02  162 
9.02  283 
9  .  02  404 
9.02525 

9.02  645 

121 
121 
I2X 
120 
X2X 

"9 
1  2O 
119 

118 
1x9 
118 
118 
117 
1x8 
116 
117 
1x6 
116 
1x6 
US 
"5 
"5 
"5 
114 
114 

"3 
114 

"3 

1X2 

1x3 

1X2 
112 
XI2 
III 
III 
III 

no 

III 

1X0 

109 

no 

109 
109 

108 
109 
108 
108 
107 
108 
107 
106 
107 
106 
106 
106 
106 
105 
105 
105 
104 

0.97838 
0.97717 
0.97596 
0-97471 
0-97355 

9  99  76i 
9-99  76o 
9-99  759 
9  99757 
9-99  756 

GO 

59 
58 

I 

.1 

.2 

•3 
•4 

•5 
.6 

•  7 
.8 

•9 

.1 
.a 
•  3 
•4 
•5 
.6 
•  7 
.8 

•9 

.3 

•3 

•4 
•  5 
.6 
•  7 
.8 

•9 

.3 

•3 
•4 

•5 
.6 

•  7 
.8 

•9 

.i 
.a 
•3 
•4 

•  5 
.6 
•  7 
.8 

•S 

.3 

•3 
•  4 
•5 
.6 
•  7 
.8 

•9 

tax 

12.  1 

34.2 

36.3 
48.4 
60.5 
73.6 
84.7 

96.8 
108.9 
»8 
xx.8 
a3.6 
35-4 
47.3 
59-0 
70.8 
82.6 

94-4 
106.3 

115 

11.5 
23.0 
34-5 
46.0 
57-5 
69.0 
80.5 
92.0 
103.5 
iia 

II.  2 
23.4 

33-6 
44-8 
56.0 
67.2 
78.4 
89.6 
100.8 

109 
10.9 

31.8 

32-7 

43.6 

54  5 
65-4 
76-3 
87.2 
98.1 
106 
10.6 

21.2 
31-8 
43-4 

53-c 

63.< 

74.3 
84.8 
95-4 

zoo 

12.0 

24.0 
3e  o 
48.0 
60.0 
72.0 
84.0 
96.o 
108.0 

"7 

"•7 

33.4 

35-1 
46.8 

58.5 
70.2 
81.9 
93-6 
105.3 

"4 

11.4 

22.8 
34.3 

45-6 
57-0 
68.4 
79.8 
91.2 

IO2.6 

III 
II.  I 

32.2 

33-3 
44-4 
55-5 
66.6 

77-7 

83.8 

99-9 
zo8 
10.8 

21.6 

3«-4 
43-2 
54-0 
64.8 
75-6 
86.4 
97.2 

zos 
10.5 

21.  0 
3»-S 
42.0 

52  S 

63.  c 
73-5 
84.  c 
94-5 

"9 

IS.  J 

*3-8  : 
,35  7 
47  6  I 
59  ! 
7i-4 
83-3 
95  -a 
107.1 

116 
li.  6  ' 
93.3 
34-8  1 
46.4  , 
58.0 
69.6 
81.3 
92.8 
104.4 

"3 
11.3 

22    6 

33  9 
45--» 
56.5 
67.8 
79-1 
90.4 
101.7 
zzo 

II.  0 
22.0 

33  o 
44-0 

55-0 
66.0 
77.0 
88.0 
99.0 
107 
10.7 
21.4 
32.1 

42.8 

53-5 
64.a 

74-9  ' 
85-6  I 
96.3 
Z04 
10.4 

20.8 

3*.  2 
41.6 
52.0 
62.4 
72.8 
83.a 
93.6 

9.02  520 
9.02639 
9.02  757 
9.02874 
9.02992 

9.02  766 
9.02885 
9.03005 
9.03  124 
9.03242 

0.97234 
0.97  115 

0.96995 
0.96  8/6 
0.96  758 

9-99755 
9-99  753 
9-99  752 
9-99751 
9-99  749 

55 
54 
53 
52 
5i 

9.03  109 
9.03  226 
9.03342 
9  03458 
9-03  574 

9.03361 
9-03479 
9-03  597 
9.03714 
9.03832 

0.96  639 
0.96  521 
o  .  96  403 
0.96  286 
0.96  1  68 

9-99  748 
9-99  747 
9-99745 
9-99  744 
9-99  742 

50 

3 
i 

II 

12 

19 

9.03690 
9.03805 
9  .  03  920 
9.04034 
9.04  149 

9.03  948 
9.04065 
9.04  181 
9.04297 
9.04413 

0.96052 

0-95  935 
0.95819 

0-95  7°3 
o  95  587 

9-99  74i 
9-99  740 
9-99  738 
9-99  737 
9-99  736 

45 
44 
43 
42 
4i 

~w 

39 
38 

3 

20 

21 
22 

23 

24 

9  .  04  262 
9.04376 
9.04490 
9.04603 
9-047I5 

9.04528 
9-04643 
9.04758 
9.04873 
9.04987 

0.95472 

0-95  357 
0.95  242 
0.95  127 
0.95013 

9-99  734 
9-99  733 
9-99731 
9-99730 
9  99  728 

g 

8 

29 

9.04828 
9.04940 
9.05052 
9.05  164 
9.05275 

9.05  101 
9.05  214 
9.05328 
9.05441 
9-05553 

0.94899 
0.94  786 
0.94672 

0-94559 
0.94447 

9  99  727 
9-99  726 
9-99  724 
9-99  723 
9  99  72i 

35 
34 
33 
32 
3i 

B 

31 
32 

33 

34 

9.05386 

9-05497 
9.05607 
9.05  717 
9.05  827 

9.05  666 
9.05  778 
9.05  890 
9.06002 
9.06  113 

0-94334 

0  .  94  222 

0.94  no 
o  .  93  998 
0.93887 

9-99  720 
9.99718 
9.99717 
9.99716 
9-99  7H 

30 

29 
28 
27 
26 

9 

11 

39 

9-°5  937 
9.06  046 
9-o6  155 
9.06264 
9.06372 

9.06  224 

9-06335 
9.06445 
9.06556 
9.06666 

0.93  776 
0.93  665 
0-93555 
0.93444 
0-93334 

9-99  713 
9  99  7" 
9.99710 

9-99  7o8 
9.99707 

25 
24 
23 

22 
21 

40 

4i 
42 
43 
44 

9.06481 
9.06  589 
9.06  696 
9  .  06  804 
9.06911 

9.06  775 
9.06885 
9.06994 
9.07  103 

9.O7  211 

0.93225 

0.93H5 
0.93  006 
0.92  897 
0.92  789 

9  99  705 
9-99  704 
9.99702 

9  99  7oi 
9.99699 

20 

19 
18 

17 
16 

* 

§ 

49 

9.07  018 
9.07  124 
9.07231 
9-07337 
9  07442 

9.07320 
9.07  428 

9  07536 
9.07643 
9.07  751 

0.92  680 
0.92572 
0.92  464 
0.92357 
0.92  249 

9.99698 
9.99696 

9-99695 
9.99693 
9.99692 

15 
14 
13 

12 
II 

50 

5i 
52 
53 
54 

9.07548 
9-07653 
9.07758 
9  07863 
9.07968 

9-07858 
9.07964 
9.08071 
9.08  177 
9.08283 

0.92  142 
0.92  036 
0.91  929 
0.91  823 
0.91  717 

9.99  690 
9.99  689 
9.99687 
9.99686 
9.99684 

10 

I 

9 

12 

59 
60" 

9.08  072 
9.08  176 
9.08280 
9.08383 
9.08486 

9.08389 
9.08495 
9.08  600 
9.08  705 
9.08810 

0.91  611 

o  9i  505 
0.91  400 
o  91  295 
o  91  190 

9.99683 
9.99681 
9.99  680 
9.99678 
9  99677 

5 
4 
3 

2 

9-08589 

9.08  914 

o  91  086 

9.99675 

0 

L.  Cos. 

(1. 

L.  Cotg. 

c.d. 

L.  Tang. 

L.  Sin. 

/ 

Prop.  Pts. 

83° 

LOGARITHMS  OF  SINE,  COSINE,  TANGENT  AND  COTANGENT,  ETC.     37 


7° 

/ 

L.  Sin. 

d. 

L.  Tang. 

c.d. 

L.  Cotg. 

L.  Cos. 

BO" 

P 

H 

Prop.  Pts. 

6 

2 

3 
4 

| 

I 

9 

9.08589 
9.08  692 

9.08795 
9.08897 
9.08999 

103 
103 

102 
102 
102 

lot 

102 

lot 
xox 

100 

tot 

TOO 
TOO 

•99 

100 

99 

99 
98 

99 
98 

98 
98 
98 
97 
97 
97 
97 
96 

97 
96 

96 
95 
96 
95 
95 
95 
94 
95 
94 
94 
93 
94 
93 
93 
93 
93 
93 
92 
93 
93 
93 

9* 

93 

9* 
9* 
9« 
9» 
90 

9' 
9° 

9.08914 
9.09019 
9  09  123 
9.09227 
9.09330 

105 
104 
104 
103 
104 
103 

«03 

103 

103 

102 

XO3 

toi 

X02 
IOX 
IOI 
XOI 
XOX 
100 
100 

too 

too 
99 
99 
99 
99 

99 

98 
98 
98 
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39 

9  22137 

9  22  21  1 
9.22286 

9  22361 
9  22435 

75 

74 
75 
75 
74 

9  22  747 
9.22824 
9.22901 
9.22977 
9  23054 

77 
77 
77 
76 

77 

o  77253 
o  77176 
0.77099 
o  77023 
o  76946 

9  99390 
9.99388 

9  99385 
9  99383 
9  99381 

25 
24 
23 

22 
21 

•4 

ii 

:I 

3< 
3 
4. 

g 

6 

J.O 

7-S 
5-0 

2-5 

).0 

2C 

3' 

* 

5 

1! 

)  6 

7.0 

f:| 

i  6 

40 

4i 
42 
1  43 

i  44 

9  22509 
9  22583 
9  22657 
9  22731 
9.22  805 

74 
74 
74 
74 

74 

9  23  130 
9.23  206 
9  23283 
9  23359 
9-23435 

76 
76 
77 
7« 
76 

0.76870 
0.76794 
0.76717 
o  76  641 

0.76565 

9  99379 
9  99377 
9  99375 
9  99372 
9  99370 

20 

19 

18 

17 
16 

•V 

.1 

.2 

•3 

D 

1. 

2 

•5 
73 

I'l 

'•9 

i 

i4 

2 

.<• 

^a 
7.2 

« 

50 

11 

;i 

49 

9.22878 
9.22952 
9  23025 
9.23098 
9.23  171 

73 
74 
73 
73 
73 

9.23510 
9  23586 
9.23661 

9  23737 
9.23812 

75 
76 
75 
76 
75 

o  76  490 
o  76414 

o  76339 
0.76263 
o  76188 

9  99368 
9  99366 
9  99  364 
9  99  362 
9  99  359 

»5 
14 
13 

12 
11 

4 

i 

II 

2< 

3< 
4. 

1 

H 

j'8 
[.i 

5-4 

•    7 

2c 

3< 

4: 

5< 

& 

.5 
).0 

5-2 

;i 

,  8 

50 

5i 
52 
53 
54 

9.23244 

9  23317 
9.23390 
9.23462 
9  23535 

73 
73 
73 
72 

73 

9-23887 
9-23962 
9.24037 

9.24  112 

9.24  186 

75 
75 
75 
75 
74 

0.76  113 
o  .  76  038 

0-75  963 

0.75888 
0.75814 

9  99357 
9  99355 
9-99353 
9  99351 
9  99348 

10 

§ 
I 

? 
i     7 

2     14 

3  21 

.       -0 

". 
i 
.1 

2 

-3 

C 

c 

C 

3 

'•3 
6 

•9 

• 
O.2 

°4 
0.6 

r»  8 

it 

12 

59 

9.23607 
9.23679 
9.23752 
9.23823 

923895 

72 
72 
73 
7i 
7* 

9.24261 

9-24335 
9.24410 

9-24484 
9.24558 

75 
74 
75 
74 

74 

0-75  739 
0.75665 
0.75590 
o.755i6 
0.75442 

9  99346 
9  99344 
9  99342 
9-9934C 
9  99337 

5 
4 
3 

2 
I 

.4    2% 

:i2 

:I3 

96? 

4 

I 
I 

n 

I 
I 

2 
2 
1 

I 

i 

4 

7 

I.O 
1.2 

Ji 

1.8 

60 

9.23967 

72 

9.24632 

74 

0.75368 

9-99335 

0 

L.  Cos. 

d. 

L.  Cot?. 

c.d. 

L.  Tang. 

L.  Sin. 

f 

r 

ro 

P« 

Pfc 

u 

80° 

40 


TABLE  IV. 


10°                                         I 

/ 

L.  Sin. 

d. 

L.  Tang. 

c.d. 

L.  Cot?. 

L.Cos. 

Prop.  Pts.         | 

0 

i 

2 

6 
4 

9-23967 
9-24039 
9.24  no 
9.24  181 
9  24253 

72 
7* 
7* 
72 

7» 

7» 
7» 
70 

7* 
70 

7» 
70 
70 
70 
70 
70 
70 
69  ' 

7<> 
69 

69 
69 
69 
69 
69 
69 
63 
69 
68 
68 
68 
68 
68 
68 
68 
67 
68 
67 
67 
67 

67 
67 
67 
67 
66 

67 
66 
67 
66 
66 
66 
66 
65 
66 
66 

65 
65 
66 
65 
65 

"dT 

9-24632 
9.24706 
9-24779 
9-24853 
9  .  24  926 

74 
73 
74 
73 
74 
73 
73 
73 
73 
73 
72 

73 
72 

73 
7* 
7« 
72 
72 
72 
7» 
7* 
7* 
73 

7* 
7» 
7* 
7« 

70 

7* 
71 
70 
7<> 
7* 
70 
70 
70 
70 

69 
70 
69 
70 
69 
69 
69 
69 
69 
69 
69 
63 
69 
68 
69 
68 
68 
68 
68 
67 
68 
68 
67 

0.75368 
0.75294 

0.75  221 

0-75  H7 
0.75074 

9-99335 
9-99333 
9-99331 
9.99328 
9.99326 

>0 

P 

11 

.1 

.2 

•3 

.4 

:! 

•9 
.1 

.2 

.3 

•A 
e 

'.I 

:l 

.9 
.1 

./ 
.1 
.( 

!s 
.9 

.1 

.2 

•3 
•4 

:1 
:i 

74 

il'i 

22.2 
29.6 

37-o 

Si 

11:1 

7« 
7-2 

14.4 

21.6 

28.8 

36.0 

43  2 
50.4 

III 

70 

7-0 
14.0 

21.0 
28.0 

35-0 
42.0 
490 
56.0 
63.0 

68 
6.8 
13-6 
20.4 
27.2 
34-0 
40.8 
47-6 
54-4 
61.2 

66 
6.6 

13.2 

19.8 
26.4 

33-c 
39-6 
46.2 
52.8 
59-4 
3 

°-; 

0.6 
0-9 

1.2 
1.5 

I.I 

2.1 

2-4 

2.7 

73 

,1:1 

21.9 

29.2 
36.5 
43-8 
Si.i 
58.4 
65-7 
71 

7-i 
14.2 
21.3 
28.4 

35-5 
42.6 

49-7 
56.6 

63-9 
69 

6.9 
13-8     ) 

20.7 
27.6 

34-5 

81 

ts 

z, 

13.4 

20.1 
26.8 

33-5 
40.2 

46.9 
53-6 
60.3 

6s 

6-5 
13.0 

195 
26.0 

32-5 
39  o 

45-5 
52.0 

58-5 

a 

O.2 

0-4 
0.6 
c  8 

I.O 
1.2 

11 

1.8 

I 
I 

& 

ii 

12 

13 
14 

9.24324 

9-2439? 
9.24466 

9-24536 
9.24607 

9.25  ooo 

9-25073 
9.25  146 

9-25219 
9-25292 

0.75  ooo 

0.74927 
0.74854 
0.74781 
0.74708 

9  99324 
9-99322 
9-993I9 
9-993I7 
9-9931? 

55 
54 
53 
52 
5i 

9.24677 
9.24748 
9.24818 
9.24888 
9.24958 

9-25365 
9-25437 
9.25  5io 
9-25  582 
9-25655 

0.74635 

0.74563 
0.74490 

0.74418 

0.74345 

9  993*3 
;  9-99  3'o 
9-99308 
9.99306 
9-99304 

50 

49 
48 
47 
46 

:* 
\i 

19 

20" 

21 
22 
23 
24 

9  .  25  028 
9-25098 
9.25  1  68 
9  25237 
9  25307 

9.25  727 

9-25  799 
9.25871 

9-25  943 
9.26  015 

0.74273 

0.74201 

0.74129 
0.74057 
0.73985 

9.99301 
9.99299 

9-99297 
9.99294 
9-99292 

45 
44 
43 
42 

41 

16" 

P 

£ 

35 
34 
33 
32 
3i 

9-25  376 
9-25445 
9.25514 
9-25583 
9  25652 

9.26086 
9.26  158 
9.26229 
9.26301 
9.26372 

0.73914 
0.73842 
0.73771 

o  73699 
0.73628 

9-99290 
9.99  288 
9.99285 
9.99283 
9.99281 

3 

3 

29 

9.25  721 
9-25  790 
9-25858 
9-25927 
9-25995 

9.26443 
9.26514 
9-26585 
9.26655 
9.26  726 

0.73557 
0.73486 

0.73415 
0.73345 

o  73  274 

9.99278 
9  99276 
9-99274 
9-99271 
9.99269 

80 

3i 

32 
33 
34 

9.26063 
9.26  131 
9.26199 
9.26  267 
9  26335 

9-26  797 
9.26867 
9.26937 
9.27008 
9.27078 

0.73203 
0.73133 
0.73063 
0.72992 
0.72  922 

9.99267 
9-99264 
9.99  262 
9.99260 
9  99257 

BO 

3 

'd 

3 

? 

* 

41 

42 
43 
44 

9  .  26  403 
9.26470 
9  26538 
9.26605 
9.26672 

9.27  148 
9.27218 
9.27288 

9-27357 
9.27427 

0.72  852 
0.72  782 
0.72  712 
0.72643 
0.72573 

9.9925? 
9.99252 
9.99250 
9.99248 
9  99245 

25 
24 

23 

22 
21 

9  .  26  806 
9.26873 
9  .  26  940 
9.27007 

9-27496 
9-27566 

9-27635 
9.27704 

9-27  773 

0.72504 
0.72434 
0.72365 
0.72  296 
0.72  227 

9  99243 
9.99241 
9.99238 
9.99236 
9  99233 

20 

!! 
\l 

.1 

.2 
,J 

-A 
c 

.6 

:l 

.9 
.1 

•4 
i 

.3 
:I 

.0 

Al 
46 

47 
48 

49 

9.27073 
9.27  140 
9.27  206 
9.27273 

9  27339 

9.27842 
9.27911 
9.27980 
9.28049 
9.28  117 

0.72  158 
0.72  089 
0.72  020 
0.71  951 
0.71  883 

9.99231 
9-99229 
9.99226 
9  99  224 
9  99  221 

15 
14 
13 

12 
II 

50 

5i 

52 

53 
J.L 

* 
9 

59 

9.27405 
9.27471 

9-27537 
9.27  602 
9.27668 

9.28  186 
9-28254 
9.28323 
9-28391 
9.28459 

0.71  814 
0.71  746 
0.71677 
0.71  609 
0.7I54I 

9.99219 
9.992I7 
9-99214 
9.99212 
9-99209 

10 

I 
I 

9-27734 
9-27799 
9.27864 
9.27930 
9-27995 
9  -*So6o 

9-28527 

9.2859! 
9.28662 
9.28730 
9.28  798 

0.7M73 
0.71405 

0.71338 
0.71  270 

0.71  202 

9.99207 
9.99204 
9.99202 
9.99200 
9-99  197 

5 
4 
3 

2 

I 

60 

9.28865 

0-71  135 

9-99  195 

0 

L*  Cos. 

L.  Cotg. 

c.  d. 

L.  Tang. 

L.  Sin. 

t 

Prop.  Pts. 

79° 

LOGARITHMS  OF  SINE,  COSINE,  TANGENT  AND  COTANGENT,  ETC.     41 


i                             11° 

f   i  iu  MIL      d. 

L.  Tang. 

c.d. 

L.  Cotg. 

L.  Cos. 

Prop.  Pts. 

0  1  9.28060 
i  1  9.28  125 

2    1    9.28  190 

3     9-28254 
4     9-28319 

65 
64 

65 
65 
64 

6S 
64 
64 

64 
63 
64 

63 
63 
64 
63 
63 
63 
63 
63 

63 
62 

63 
62 
62 
63 
6a 
62 
61 
6a 
62 
61 
62 
61 
6a 
61 
61 
61 
61 
61 
61 
60 
61 
60 
61 
60 
61 
60 
60 
60 
60 

59 
60 
60 

59 
60 

9.28865 

9-28933 
9.29000 
9.29067 
9.29  134 

68 
67 
67 
67 
67 
67 
67 
67 
66 
67 
66 

67 
66 
66 
66 
66 
66 
66 
66 

65 
65 

65 
65 
65 
65 

64 

65 
64 
65 
64 

65 
64 
64 
64 
64 
63 
64 
63 
64 
63 
64 
63 
63 
63 
63 
63 
63 
63 
62 

62 

63 
62 

62 

62 

0.71  i35 
0.71  067 
0.71  ooo 
0.70933 
0.70  866 

9-99  195 
9-99  192 
9-99  190 
9-99  187 
9-99  185 

GO 

59 

58 

g 

.2 

•  3 
-4 

:I 
i 

•9 
.1 

.2 

•3 
•4 

•9 
.1 

.2 

.3 

•4 
.  c 

6 

k 

.9 
.1 

•A 
\l 

.9 
.1 

,4 

1  1 

.6 

;j 

.i 
.1 

.6 

t 

:l 
.9 

68 

6.8 

20.4 
27.2 
34-0 
40.8 
47-6 
54-4 
61.2 

66 

6.6 
13.2 
10.8 

26.4 
33-o 
39-6 
46.2 
52.8 
59-4 

6.4 

12.8 

19.2 

32.0 
38.4 
44-8 

57^6 
6a 
6.2 

12  ./. 

18.6 
24.8 
31.0 
37-2 
43-4 
49-6 
55-8 
60 
6.0 

12.0 

18.0 
24.0 
30.0 
36.0 
42.0 
48.0 
54-0 
3 

0.9 

1.2 

1:1 

2.1 

2.4 

2.7 

6? 
6.7 

I3.4 
20.1 
26.8 

33-5 
40.2 
46.9 
53-6 
60.3 

65 

6-5 
13-0 
195 
26.0 

32.5 
39-o 

45-5 
52.0 

58.5 
63 
6-3 

12.6 

18.9 
25.2 

44.1 
50.4 
56.7 
6s 
6.1 

12.2 

18-3 
24.4 

36i 
42.7 
48.8 
54-9 

59 

5-9 
11.  8 

17.7 
23-6 
29-5 
35-4 
41-3 
47.2 

53-1 
a 
0.2 

o!s 

I.O 
1.2 

i.B 

7 

9 

"tin 

12 
13 

14 

9-28384 
9.28448 
9.28512 
9.28577 

9  .  28  641 
9  .  28  705 
9.28769 
9-28833 
9.28896 
9.28960 

9.29  201 
9.29268 

9-29335 
9.29402 

9  .  29  468 

0.70799 
0.70  732 
0.70665 
0.70598 
0.70532 

9.99  182 
9.99  180 
9-99  177 
9-99  175 
9-99  172 

55 
54 
53 
52 
_51_ 

3 

47 
46 

9-29535 
9.29  601 
9.29668 

9-29  734 
9.29  800 

0.70465 
0.70399 
0.70332 
0.70  266 

0.70  200 

9-99  170 
9.99  167 
9-99  165 
9.99  162 
9.99  160 

15    1    9.29024 

16  I  9.29087 
17     9.29  150 
18  1  9.29214 
19  1  9.29277 

9.29866 
9.29932 
9-29998 
9.30064 
9.30  130 

0.70  134 
0.70068 
0.70002 
0.69936 
0.69  870 

9-99  157 
9-99  155 
9-99  152 
9.99150 

9  99  H7 

45 
44 
43 
42 

41 

1T 

P 

i 

20 

21 
22 
23 

24 

9.29340 
9.29403 
9.29466 

9  29  529 
9.29591 

9-30I95 
9.30261 
9.30326 
9-30391 
9-30457 

0.69  805 

0-69  739 
0.69674 
0.69609 
0.69  543 

9  99H5 
9-99  '42 
9-99  HO 
9-99  137 
9-99  135 

27 
29 

30 

32 

P 

9-29654 
9.29716 
9.29  779 
9.29841? 
9-29903 
9  .  29  966 
9  .  30  028 
9.30090 
9  30151 
9  30213 

9.30275 
9-30336 
9-30398 

9.30521 

9-30522 
9-30587 
9.30652 
9.30717 
9.30782 

0.69478 
0.69413 
0.69348 
0.69  283 
0.69  218 

9-99  132 
9  99  13° 
9  99  127 
9-99  124 
9.99122 

35 
34 
33 
32 

9.30846 
9.30911 

9-30975 
9.31  040 
9.31  104 

0.69  154 
0.69  089 
o  .  69  025 
0.68960 
0.68896 

9.99119 
9.99117 

9  99  "4 
9.99  112 
9  99  109 

30 

11 

11 

f 

s 

44 

9.31  168 

9  -31  233 
9.31  297 
9.31  361 

0.68832 
0.68  767 
0.68  703 
0.68639 
0.68575 

9.99  106 
9-99  104 
9.99  101 
9.99099 
9.99096 

25 
24 

23 

22 
21 

9.30582 

9-30643 
9.30  704 

9-30765 
9  .  30  826 

9.31489 

9-31  55| 
9.31  616 
9.31  679 
9-3i  743 

0.68511 
0.68448 
0.68384 
0.68321 
0.68257 

9.99093 
9.99091 
9.99088 
9.99086 
9.99083 

20 

5 

|45 
46 
47 

|48 

i  49 

9-30887 

9-30947 
9.31  008 
9.31  068 
9.31  129 

9.31806 
9.31  870 

9-31  933 
9.31  996 

9-32059 

0.68  194 
0.6*8  130 
0.68067 
0.68004 
0.67941 

9  .  99  080 
9.99078 

9-99075 
9-99072 
9.99070 

i5 
13 

12 
II 

150 

52 

53 
54 

9 

II 

!  59_ 

9.31  189 
9  31  250 
9.31310 
9-31  370 
9-31  430 

9.32  122 
9-32  185 
9-32248 
9-32311 
9-32373 

0.67878 
0.67815 
0.67  752 
0.67  689 
0.67  627 

9.99067 
9.99064 
9  .  99  062 

9.99059 
9-99056 

10 

I 

I 

9.31  490 

9-31  549 
9.31  609 
9.31  669 
9.31  728 

9-32436 
9.32498 
9.32561 
9.32623 
9.32685 

0.67  564 
0.67  502 
0.67439 
0.67377 
0.67315 

9-99054 
9.99051 
9.99048 
9-99046 
9.99043 

5 
4 
3 

2 

I 

9.31  788 

9-32  747 

0.67253 

9.99040 

0 

L.  Cos. 

d. 

L.  Cotg.  c.  d. 

L.  Tans. 

L.  Sin. 

t 

Prop.  Pts. 

78° 

TABLE  IV. 


12° 

t 

L.  Sin. 

d. 

L.  Tang. 

c.d. 

L.  Cotg. 

L.  Cos. 

p 

rop.  ] 

Pts. 

0 

I 

2 

3 
4 

9.31  788 

9-31  847 
9.31  907 
9.31  966 
9.32025 

59 
60 

59 

59 

rg 

9-32747 
9.32810 
9.32872 
9.32933 
9-32995 

63 
62 

61 
62 
62 

0.67253 
0.67  190 
0.67  128 
0.67067 
0.67  005 

9.99040 
9.99038 

9-99035 
9.99032 
9.99030 

(JO 

3 

11 

.1 

.2 

•  3 

63 
6.3 

12.6 

18.9 

62 

6.2 

12  4 
18.6 

I 
I 

9 

9.32084 

9-32  H3 
9.32202 
9.32261 
9-32  3*9 

59 
59 
59 
58 

9-33057 
9-33  "9 
9-33  *  80 
9-33242 
9.33303 

62 
61 
62 
61 
62 

o  .  66  943 
0.66881 
0.66  820 
0.66758 
0.66697 

9.99027 
9.99024 
9  .  99  022 
9.99019 
9.99  016 

55 
54 
53 
S2 
5i 

•4 

:f 
•I 

25.2 

3J:i 

44.1 
50.4 

24.8 
31.0 

37-2 
43-4 
49-6 

10 

ii 

12 
13 

14 

9-32378 
9.32437 
9-32495 
9-32553 
9.32  612 

59 
58 
58 
59 

eg 

9.33365 
9.33426 
9.33487 
9-33548 
9-33609 

61 
61 
61 
61 

61 

0.66635 
0.66574 
0.66  513 
0.66452 
0.66391 

9.99013 
9.99011 
9.99008 
9.99005 
9.99002 

50 

3 

s 

•9 

.2 

•3 

i>6-'; 

61 

6.1 

12.2 
18-3 

55-8 
60 
6.0 

12.0 

18.0 

II 

ii 

19 

9-32670 
9.32  728 
9-32  786 
9-32844 
9.32902 

58 
58 
58 
5» 

t-8 

9.33670 
9-33  731 
9-33  792 
9.33853 
9.33913 

61 
61 
61 

60 
61 

0.66  330 
0.66  269 
0.66  208 
0.66  147 
0.66087 

9.99  ooo 

9.98997 
9.98994 

9.98991 
9  .  98  989 

45 
44 
43 
42 
41 

•4 

ii 

24.4 

30-5 
36.6 

42-7 
48.8 

24.0 
30.0 
36.0 
42.0 
48.0 

20 

21 
22 

\   23 
24 

9.32960 
9.33018 
9-33075 
9-33  133 
9-33  190 

58 
57 
58 

57 
eg 

9-33974 
9-34034 
9-34095 
9.34155 
9.34215 

60 
61 
60 
60 
6z 

0.66  026 
0.65  966 
0.65  905 
0.65  845 
0.65  785 

9.98  986 
9.98983 
9.98980 
9.98978 
9-98975 

40 

3 

11 

•9 

54-9 

5 

.i     5 

.2    II 
•3    IJ 

54-° 

9 

'I 

25 
26 

3 

29 

9-33248 
9-33305 
9-33362 
9-33420 
9-33477 

57 
57 
58 
57 

9.34276 
9.34336 
9.34396 
9.34456 
9-345I6 

60 
60 
60 
60 
60 

0.65  724 
0.65  664 
0.65  604 
0.65  544 
0.65  484 

9-98972 
9.98969 
9.98967 

9-98964 
9.98961 

35 
34 
33 
32 
31 

•4  *J 
.5  29 
•6  35 
.7  4i 
.8  45 

Q    H^ 

.6 
•5 
•4 
-3 

.2 

IT 

30 

3i 
32 
33 
34 

9-33  534 
9-33591 
9-33647 
9-33  704 
9-3376I 

57 
56 
57 
57 

9.34576 
9.34635 
9.34695 
9-34755 
9.34814 

59 
60 
60 

59 
60 

0.65  424 
0.65  365 
0.65  305 
0.65  245 
0.65  186 

9.98958 
9.98955 
9.98953 
9.98950 
9.98947 

30 

3 
Z 

.1 

.2 

•3 

•y  jj 
58 

5-8 
n.  6 

17-4 

57 

5-7 
11.4 

17.1 

•it     O 

35 
36 

% 

39 

9.338i8 
9-33  874 
9-33931 
9-33987 
9-34043 

57 
56 
57 
56 
56 

9-34874 
9-34933 
9-34992 
9-3505I 
9-35IH 

59 
59 
59 
60 

0.65  126 
0.65  067 
0.65  008 
0.64  949 
0.64889 

9.98944 
9.98941 
9.98938 
9.98936 
9-98933 

25 
24 

23 

22 
21 

ii 
ii 

.0 

23.2 
29.0 
34-8 
40.6 

46.4 

tJ2.2 

28.5 

34-2 
39-9 
45-6 
51.3 

40 

4i 

42 

43 
44 

9-34  loo 
9.34I56 
9.34212 
9.34268 
9-34324 

56 
5« 

56 
56 

efi 

9-35  170 
9-35  229 
9-35  288 
9-35347 
9-35  405 

59 
59 
59 
58 

o  .  64  830 
0.64  771 
0.64  712 
0.64653 
0.64595 

9.98930 
9.98927 
9.98924 
9.98  921 
9.98919 

20 

ii 

11 

.1 

.2 

•3 

56 

5-6 

II.  2 

16.8 

122  4 

55 

5-5 

II.  0 

16.5 

22  O 

$ 
9 

49 

9-3438o 
9-34436 
9-34491 
9-34547 
9.34602 

5° 
56 

55 
56 
55 

9.35464 
9.35  5J3 
9-35  58i 
9-35  640 
9-35698 

59 
58 
59 
58 

0.64536 
0.64477 
0.64419 
0.64  360 
0.64  302 

9.98  916 
9.98913 
9.98910 
9.98907 
9.98904 

15 
14 
13 

12 

II 

ii 

.9 

28.0 
33-6 

39-2 
44-8 
50.4 

27-5 

33-0 
38-5 
44-0 
49-5 

50 

5i 
52 
53 

54 

9-34658 
9-347I3 
9-34769 
9-34824 
9-34879 

5° 
55 
56 
55 
55 

9-35  757 
9-35815 
9-35  873 
9-35931 
9-35  989 

58 
58 
58 

58 

eg 

0.64243 
0.64  185 
0.64  127 
0.64069 
0.64  on 

9.98  901 
9.98898 
9.98896 
9.98893 
9.98890 

10 

I 
I 

.1 

.2 
4 

3 

°-3 
0.6 

0.9 

1.2 

a 
10.2 
0.4 
0.6 
0.8 

9 

9 

1  59 

9-34934 
9.34989 
9-35044 
9.35099 
9-35  154 

55 
55 
55 
55 
55 

9.36047 
9.36  105 

9-36  163 
9.36221 
9.36279 

58 
58 
58 
58 

0.63953 
0.63  895 
0.63  837 
0.63  779 
0.63  721 

9.98887 
9.98884 
9.98881 
9.98  878 
9.98875 

5 
4 
3 

2 
I 

I 

I 

i 

4 

ii 

2.1 

2.4 
2.7 

I.O 
1.2 

H 

1.8 

60 

9-35209 

55 

9.36336 

0.63  664 

9.98  872 

0 

L.  Cos. 

d. 

L.  Cotg. 

c.d. 

L.  Tang. 

L.  Sin. 

t 

I 

>rop. 

Pts. 

77° 

LOGARITHMS  OF  SINE,  COSINE,  TANGENT  AND  COTANGENT,  ETC.     43 


1  3°                                       I 

I  , 

L.  Sin. 

d. 

L.  Tang. 

c.d. 

L.  Cotg. 

L.  Cos. 

Prop.  Pts. 

0 

9-35209 

9-36336 

eg 

0.63  664 

9.98872 

60 

i 

9-35  263 

9.36394 

eg 

o  63  606 

9.98869 

59 

58 

57 

2 

9-35  3l8 

55 

9-36452 

0.63  548 

9.98867 

S8 

.1 

S-8 

5-7 

3 

9-35373 

55 

9-36509 

0.63491 

9.98864 

57 

.2 

11.6 

II.4 

4 

9-35427 

54 

9.36566 

58 

0.63434 

9.98861 

56 

•3 

17.4 

I7.I 

I 

9-3548i 
9-35  S36 

55 

9.36624 
9.36681 

57 

0.63  376 
0.63319 

9.98858 
9-98855 

55 
S4 

4 

23.2 
29.0 

22.8 
28.5 

7 

9-35590 

54 

9-36738 

57 

0.63  262 

9.98852 

53 

.6 

34-8 

34-2 

8 

9-35  644 

54 

9.36  795 

57 

0.63  205 

9.98849 

S2 

•7 

40.6 

39-9 

9 

9-35698 

54 
54 

9.36852 

57 

57 

0.63  148 

9.98846 

51 

.8 

46.4 

45-6 

10 

9-35  752 

9-36909 

0.63091 

9-98843 

50 

•9 

52.2 

5^3 

ii 

9.358o6 

54 

9.36966 

57 

0.63  034 

9.98  840 

49 

56 

55 

12 
13 
H 

9-35  860 
9-359H 

54 
54 

9.37023 
9.37080 
9-37  137 

57 

57 

eg 

0.62977 
0.62  920 
0.62863 

9-98837 
9.98834 
9-98831 

48 
47 
46 

.2 

-3 

5-6 

II.  2 

16.8 

5.5 

II.  0 

16.5 

11 

17 

9.36  022 
9-36075 
9  36  129 

53 
54 

9-37  193 
9-37250 
9.37306 

57 

56 

0.62  807 
0.62  750 
0.62694 

9.98828 
9.98825 
9.98822 

45 
44 
43 

•4 

22.4 
28.0 
33-6 

22.0 
27-5 

33-o 

18 

9.36  182 

53 

9-37363 

57 

0.62637 

9.98  ^19 

42 

•I 

39-2 

38-5 

19 

9-36236 

54 
53 

9.37419 

5° 

0.62  581 

9.98816 

.8 

44.8 

44-0 

20 

9.36289 

9-37476 

0.62524 

9-988i3 

40 

•9 

5°'4 

49-5 

21 

9-36342 

53 

9-37532 

e£ 

0.62468 

9.98  810 

39 

54 

22 
23 

24 

9-36395 
9.36449 
9-36502 

53 
54 
53 

CO 

9.37588 
9-37644 
9-37  700 

5° 

56 
56 

0.62412 
0.62  356 
0.62  300 

9.93  807 
9.98804 
9.98801 

38 

11 

.1     5-4 

.2    10.8 

•3   16.2 

% 

9.36555 
9.36608 

53 

9.37756 
9.37812 

56 

0.62244 
0.62  188 

9.98  798 
9  98795 

35 
34 

.4  21.6 

.5  27.0 

27 

9.36660 

5~ 

9.37868 

50 

0.62  132 

9.98  792 

33 

.6  32.4 

29 

9.36713 
9.36766 

53 
53 
51 

9-37924 
9.37980 

5° 
56 

0.62076 
o  .  62  020 

9.98  789 
9.98786 

32 

'I  37-8 
.8  43-2 

9..Q    6 

30 

9.36  819 

9-38035 

0.61  965 

9-98783 

80 

40.0 

9.36871 

52 

9.38091 

56 

0.61  909 

9.98780 

29 

53 

S* 

32 
33 
34 

9.36924 
9.36976 
9.37028 

53 
52 

52 

9.38  147 
9.38202 
9-38257 

50 
55 
55 

0.61  853 
0.61  798 
0.61  743 

9.98  777 
9.98  774 
9.98771 

28 

11 

.1 

.2 

•3 

5-3 
10.6 

15-9 

5-2 
10.4 

15.6 

1 

39 

9-37o8i 
9-37I33 
9.371S5 

9  37237 
9-37289 

52 

52 

52 
52 

9-383I3 
9.38368 

9-38423 
9-38479 
9.38534 

55 

55 
56 
55 

0.61  687 
0.61  632 
0.61  577 
o.Ci  521 
0.61  466 

9.98768 
9.98  765 
9.98  762 

9-98  759 
9.98  756 

25 
24 
23 

22 
21 

•4 

•  7 

21  .2 
26.5 

3?  8 

37-i 
42.4 

20.  0 

26.0 
31.7 

36.4 
41.6 
46  8 

40 

9-37341 
9-37393 

52 

9-38589 
9.38644 

55 

55 

0.61  411 
0.61  356 

9.98  753 
9.98  750 

20 

19 

. 

51 

4 

42 

9-37445 

52 

9.38699 

55 

o.Ci  301 

9.98  746 

18 

.1 

b-1 

0.4 

43 

44 

9-37497 
9-37549 

S2 
52 

9!  38  808 

55 
54 

0.61  246 
0.61  192 

9-98  743 
9.98740 

\l 

.2 

•3 

10.2 
15-3 

1.2 

45 
46 

9.37600 

52 

9.38863 
9.38918 

55 
55 

0.61  137 
0.61  082 

9-98  737 
9-98  734 

15 

14 

•4 

25-5 

2.0 

11 

9-37703 
9-37755 

Si 
52 

9-38972 
9.39027 

54 
55 

0.61  028 
0.60973 

9.98  73i 
9.98  728 

13 

12 

:J 

Hi 

49 

9.37806 

51 

9.39082 

55 

0.60918 

9-98  725 

II 

O 

AC    Q 

\  6 

50 

9.37858 

9-39  136 

54 

0.60864 

9.98  722 

10 

51 

9-37909 

5X 

9.39  190 

54 

0.60810 

9.98  719 

9 

3 

52 
53 

54 

9.37960 
9.38011 
9.38062 

5* 
5» 

9.39245 
9.39299 

9-39353 

55 
54 
54 

0.60  755 
0.60  701 
0.60647 

9.98  715 
9.98  712 
9.98  709 

I 

.1 

.2 
-3 

0.3 

0.6 
0.9 

I* 

O.2 
O.O 

o  8 

P 

9.38  "3 
9.38  164 

S« 

9.39407 
9.39461 

54 
54 

0.60593 
0.60539 

9.98  706 
9.98  703 

5 

4 

1:1 

1.0 
1.2 

9-38215 
9.38266 

5X 
5« 

9.39515 

54 

54 

0.6048? 
0.60431 

9.98700 
9.98697 

3 

i 

2.1 

2  A 

59 

9-38317 

5« 

9-39623 

54 

0.60377 

9.98694 

i 

.9 

if  .4 

2.7 

1.8 

60 

9-38368 

9-39677 

54 

0.60323 

9  .  98  690 

0 

L.  Cos. 

d. 

L.  Cotg. 

c.d. 

L.  Tang. 

L.  Sin. 

t 

Prop.  Pts. 

76° 

44 


TABLE  IV. 


14° 

t 

L.  Sin. 

d. 

L.  Tang. 

.d. 

L.  Coter. 

L.  Cos. 

d. 

T] 

rop.  ] 

>ts. 

1  ° 

I 

2 

3 

4 

9-38368 
9.38418 
9.38469 
9-38519 

9-38570 

so 
51 
50 
51 

5° 

9.39677 
9-39  73i 
9.39785 
9-39838 
9-39892 

54 
54 
53 
54 
53 

0.60323 
o  .  60  269 
0.60  215 
0.60  162 
0.60  108 

9.98690 
9.98687 
9.98684 
9.98681 
9.98678 

3 
3 
3 
3 

00 

59 
58 

y 

.1 

2 

54 

5-4 
10  8 

53 
IO  O 

1 

I 

9 

9.38620 
9.38670 
9-38721 
9.38  771 
9.38821 

So 
Si 
So 
5° 
SO 

9-39945 
9-39999 
9.40052 
9.40  106 
9-40  159 

54 
53 
54 
53 
53 

0.60055 
0.60001 
0.59948 
0.59894 
0.59841 

9-98675 
9.98671 
9.98668 
9.98665 
9.98662 

4 
3 
3 
3 

55 
54 
53 
52 
5i 

•3 
-4 

.7 

16.2 

21.6 

27.0 

SI 

15-9 

21.2 
26.q 
318 

37-1 

10 

ii 

12 
13 

H 

9.38871 
9.38921 
9.38971 
9.39021 
9.39071 

So 
So 
So 
So 

9.40212 
9.40266 
9.40319 
9.40372 
9.40425 

54 
53 
53 
53 

0.59788 

0-59734 
0.59681 
0.59  628 
0-59575 

9.98659 
9.98  656 
9-98652 
9.98649 
9  .  98  646 

3 
4 
3 
.3 

60 

3 

% 

.8 
•9 

&6 
5* 

42-4 
47-7 

5« 

\l 

11 

19 

9-39  121 
9  39  170 
9.39220 
9.39270 
9.393I9 

49 
50 
50 
49 

9.40478 

9-40531 
9.40584 
9.40  636 
9.40689 

53 
53 
52 
53 
53 

0.59522 
0.59469 
0.59416 
0.59364 
0-593" 

9-98643 
9.98640 
9.98636 
9-98633 
9.98630 

3 

4 
3 
3 

45 
44 
43 
42 
41 

.1 

.2 

•3 
•4 

5-2 
10.4 
15.6 
20.8 

26.0 

5-x 

IO.2 

15.3 
20.4 

25-5 

20 

21 
22 

23 

24 

9-39369 
9.39418 

9-39467 
9.395I7 
9.39566 

49 
49 
So 
49 

9.40  742 

9-40  795 
9.40847 
9.40900 
9.40952 

53 

53 
53 
53 

0.59258 
0.59205 

0.59153 
0.59  100 
0.59048 

9.98627 
9.98  623 
9  98  620 
9.98617 
9.98  614 

4 
3 
3 
3 

40 

39 
38 

11 

i 

•  9 

31.2 

36.4 

41.6 

46.8 

30.0 

35-7 
40.6 

45-9 

> 

8 

3 

29 

9-396i5 
9.39664 

9-397I3 
9.39762 
9.39811 

49 
49 
49 
49 

9-41  005 
9-4i  057 
9.41  109 
9.41  161 
9.41  214 

53 
53 
53 
53 

52 

0.58995 
0.58943 
0.58891 
0.58839 
0.58786 

9.98  610 
9.98607 
9.98604 
9.98601 
9-98597 

3 
3 
3 

4 

35 
34 
33 
32 
3i 

.1 

2 

•3 

A 

50 
5-0 

10.  0 

15.0 

20  o 

49 

4-9 

9.1 

14.7 

IQ.6 

80 

3i 
32 
33 
34 

9.39860 
9.39909 

9-3995? 
9.40006 
9-40055 

49 
49 
48 
49 
48 

9.41  266 
9.41  318 
9.41  370 
9.41  422 
9.41  474 

53 
53 
53 
53 
5a 

0.58  734 
0.58682 
0.58630 
0.58578 
0.58  526 

9.98594 
9.98591 
9-98588 
9.98584 
9.98581 

3 
3 

4 
3 

80 

3 
2 

:! 

.9 

25.0 
30.0 

35.0 
40.0 
45.0 

24.5 
29.4 

34-3 
39.2 
44.1 

P 

H 

39 

9-40  103 
9-40  152 
9.40200 
9.40249 
9.40297 

49 
48 
49 
48 

9-41  526 
9.41  578 
9.41  629 
9.41  68  i 
9-41  733 

5« 

5» 

5« 

53 

0.58474 
0.58422 
0.58371 
0.58319 
0.58267 

9.98578 
9.98574 
9-9857I 
9-98568 
9-98565 

4 
3 
3 
3 

25 

24 

23 

22 
21 

.1 

.2 

48 

4-8 
9.6 

47 

4-7 
9.4 

40 

4i 

42 

43 

44 

9.40346 
9  40394 
9-40442 
9.40490 
9-40538 

48 

48 
48 

48 

Aa 

9.41  784 
9.41  836 
9.41  887 
9.4I939 
9.41  990 

5* 
Si 
53 

S» 

0.58  216 
0.58  164 
0.58  113 
0.58061 
0.58010 

9.98  561 
9.98558 
9.98555 
9.98551 
9-98548 

3 
3 

4 
3 

20 

18 

II 

•3 
•  4 

14.4 
19.2 

24.0 
28.8 
3,V6 

ls',8 

S:ll 
32.9  1 

3 

s 

49 

9.40586 
9.40634 
9.40682 

9.4073° 
9.40  778 

48 
48 
48 
48 

9.42041 
9.42093 
9.42  144 
9.42  195 
9.42  246 

53 
51 
Si 
5* 

0-57959 
0.57907 
0.57856 
0.57805 
0-57754 

9-98545 
9.98541 
9.98538 
9.98535 
9-98531 

4 
3 
3 
4 

15 

14 
13 

12 
II 

•9 

38.4 
43  2 

4 

37.6 

: 

sr 

!  5I 

1    C2 

1  53 
54 

9.40825 
9.40873 
9.40921 
9  40  968 
9.41  016 

47 
48 
48 
47 
48 

9.42297 
9-42348 
9.42399 
9-42450 
9.42501 

5i 
5» 
Si 
5» 

0-57  703 
0.57652 
0.57601 
0.57550 
0.57499 

9.98528 

998525 
9  98521 
9-98518 
9  985*5 

3 
3 
4 
3 
3 

10 

1 

.1 

.2 

•3 
•4 

0.4 

o.S 

1.2 

1.6 

2.C 

0.3 

0.6 
0.9 

1.2 

;-j 

55^ 

? 

59 

9.41063 
9.41  in 
9.41  158 
9.41  205 
9  41  252 

48 
47 
47 
47 

.0 

9.42552 
9-42603 
9-42653 
9-42  704 
9-42  755 

5* 

50 
S» 
Si 

0.57448 
0-57397 
0-57347 
0.57296 
0.57245 

9.98511 
9-98508 

9-98505 
9.98501 

9.98498 

4 

3 
3 

4 
3 

5 
4 
3 

2 
I 

:l 

•  9 

2.4 
2.8 

3:! 

1.8 

2.1 

2-4 
2-7 

it  jo 

9.41  300 

9.42805 

0-57  195 

9.98494 

0 

L.  Cos. 

d. 

L.  Cotg. 

c.d 

L.  Tang 

L.  Sin. 

d. 

t 

1 

*rop. 

Pts. 

75° 

LOGARITHMS  OF  SINE,  CO.Bl^,  TANGENT  AND  COTANGENT,  ETC. 


45 


15° 

t 

L.  Sin. 

d. 

L.  Tang. 

c.d. 

L.  Cotg. 

L.  Cos. 

d. 

Prop.  Pts. 

0 

9.41  300 

9.42805 

O-57  J95 

9.98494 

00 

I 

9-4i  347 

9-42856 

0.57  144 

9.98491 

S9 

2 

9-41  394 

9.42906 

0.57094 

9.98488 

3 

S8 

SI 

5° 

3 
4 

9.41441 
9.41  488 

47 
47 

9-42957 
9.43007 

50 
50 

0.57043 
0-56993 

9.98484 
9.98481 

3 
4 

H 

.1 

.2 

IO.2 

5-o 

IO.O 

i 

I 

9 

9-4i  535 
9.41  582 
9.41  628 
9-4i  675 
9.41  722 

47 
46 

47 
47 
46 

9-43057 
9-43  108 
9-43  158 
9.43208 
9-43258 

50 
50 
50 

5° 

0-56943 
0.56  892 
0.56  842 
0.56  792 
0.56  742 

9.98477 
9.98474 
9.98471 
9.98467 
9.98464 

3 
3 

4 
3 

55 
54 
53 
52 

.3 

•4 

15-3 
20.4 

25-5 
30.6 

3S-7 

15-0 

20.  o 

25-0  ; 
30.0 

35-o 

10 

9.41  768 

9-43308 

0.56692 

9  .  98  460 

50 

.8 

40.8 

40.0 

12 

9.41815 
9.41  861 

47 
46 

9-43358 
9.43408 

5° 
50 

0.56642 
0.56592 

9-98457 
9-98453 

3 

4 

49 
48 

•9 

45-9 

45-0 

13 

14 

9.41  908 
9-41  954 

47 
46 

9-43458 
9-43508 

5° 
50 
5° 

0.56542 
0.56492 

9.98450 
9.98447 

3- 
3 

47 
46 

49 

48 

ii 

9.42001 
9.42047 
9.42093 
9.42  140 

46 
46 
47 

9.43558 
9.43607 

9.43657 
9-43  707 

49 
50 
50 

0.56442 
0.56393 
0.56343 
0.56293 

9-98443 
9-98440 
9.98436 

9-98433 

3 

4 
3 

45 
44 
43 
42 

.1 

.2 

•3 
•4 

49 
9-8 

14.7 
19.6 

4-8 
9-6 
14.4 
19.2 

19 

9.42  186 

40 
46 

9-43756 

49 
5° 

0.56244 

9.98429 

4 

41 

•5 

24-5 

21 
22 
23 

9.42232 
9.42278 
9.42324 
9.42370 

46 
46 
46 
.f. 

9-43855 
9-43954 

49 
50 
49 

0.56  194 
0.56  145 
0.56095 
0.56046 

9.98  426 
9.98422 
9.98419 
9.98415 

4 
3 
4 

40 

39 
38 
37 

•9 

29.4 

34.3 
39-2 
44.1 

pU 

43-2 

1   24 

9.42416 

AC 

9.44004 

40 

0.55996 

9.98412 

3 

36 

f 

? 

29 

9.42461 
9.42507 
9-42553 
9.42599 
9.42644 

46 
46 
46 

45 
46 

9-44053 

9-44  102 

9.44I5I 
9.44201 
9.44250 

49 
49 
So 
49 

0-55947 
0.55898 
0.55849 
0-55  799 
0.55750 

9.98409 
9.98405 
9.98402 
9.98398 
9-98395 

4 
3 
4 
3 

35 
34 
33 
32 
31 

.1 

.2 

•3 

A 

47 

4-7 
9-4 

14.1 
18  8 

4.6 
9-2 
n.8 
18.4 

30 

9.42690 
9.42735 

45 

.e. 

9.44299 

49 

0.55  701 
0-55652 

9.98391 
9.98388 

3 

29 

23.5 
28.2 

23.0 
27.6 

32 
33 
34 

9.42  781 
9.42  826 
9.42872 

40 
45 
46 

45 

9-44397 
9.44446 
9.44495 

49 
49 
49 

0.55603 
0-55554 
0.55505 

9-98384 
9.98381 

9.98377 

4 
3 

4 

28 
11 

.9 

32.9 
37-6 

32.2 
36.8 
41.4 

P 

9.42917 
9.42962 

45 
Af. 

9-44544 
9-44592 

48 

0.55456 
0.55408 

9-98373 
9.98370 

3 

25 
24 

ii 

39 

9.43008 

9.43053 
9.43098 

4° 
45 
45 

9.44641 
9-44690 
9.44738 

49 
49 
48 

0-55359 
0.55310 
0.55  262 

9-98366 
9-98363 
9.98359 

4 
3 

4 

23 

22 
21 

2 

45 

4-5 
9O 

44 

it 

40 

9-43  143 

9.44787 

49 

0-55213 

9.98356 

3 

20 

.3 

41 

9-43  l88 

45 

9.44836 

49 

0-55  164 

9.98  352 

4 

19 

•  4 

18.0 

17  6 

42 
43 
44 

9.43233 
9.43278 

9.43323 

45 

45 
45 

9-44884 
9-44933 
9.44981 

48 
49 
48 

A9 

0.55  116 
0.55067 
0.55019 

9-98349 
9-98345 
9.98342 

3 

4 
3 

\l 

22.5 
27.0 

v-s 

22.  0 
26.4 
30.8 

? 

9.43367 
9.43412 

9-43457 

45 
45 

9-45029 
9-45078 
9-45  126 

49 
48 

0-54971 
0.54922 
0.54874 

9-98338 
9-98334 
9-9833I 

4 

4 
3 

15 
H 
13 

•9 

36.0 
40.5 

35-2 
39-6 

48 

9-43502 

45 

9-45  174 

48 

0.54826 

9-98327 

4 

12 

49 

9-43546 

44 

9-45222 

48 

0.54778 

9.98324 

3 

II 

4 

3 

50 

9-43591 

9-45  271 

49 

0.54729 

9.98320 

4 

10 

.1 

04 

03 

5i 

52 

9-43635 
9.43680 

44 
45 

9-45  319 
9-45  367 

48 
48 

0.54681 
0.54633 

9.98317 
9-983I3 

3 

4 

1 

.2 
•3 

0.8 
1.2 

0.6 
0.9 

53 

9-43  724 

44 

9.45  4i5 

48 

0-54585 

9.98309 

4 

7 

•4 

1.6 

1.2 

9-43  769 

45 

9-45463 

48 

.0 

0-54537 

9.98306 

3 

6 

2.O 

1-j 

55 

9-43813 

9-455" 

4° 

0.54489 

9.98302 

4 

5 

.b 

2.4 

1.8 

5<> 

9-43857 

44 

9-45559 

48 

0.54441 

9.98299 

3 

4 

•  7 

2.8 

2.1 

57 

9.43901 

44 

9-45  606 

47 

0-54394 

9.98295 

4 

3 

.8 

3.2 

2.4 

58 

9-43  946 

45 

9-45  654 

48 

0.54346 

9.98291 

4 

2 

•  9 

3-6 

2.7 

59 

9-43990 

44 

9.4^702 

48 

0.54298 

9.98288 

3 

I 

GO 

9-44034 

9-45  750 

4° 

o  54  250 

9.98284 

4 

0 

L.  Cos. 

d. 

L.  Cotg. 

c.d. 

L.  Tang. 

L.  Sin. 

d. 

, 

Prop.  P^. 

74° 

TABLE  IV. 


16° 

/ 

L.  Sin. 

d. 

L.  Tan?. 

c.d. 

L.  Cot?. 

L,  Cos. 

d. 

p 

rop.  1 

Pis. 

0 

I 

2 

3 
4 

9.44034 
9.44078 

9.44  122 

9.44  166 
9.44210 

44 
44 
44 
44 
43 

9-45  750 
9-45  797 
9.4584? 
9-45892 
9-45  940 

47 
48 
47 
48 
47 

0.54250 
0.54203 

0.54155 
0.54  1  08 
0.54060 

9  .  98  284 
9.98281 
9.98277 
9.98273 
9.98270 

3 
4 

4 
3 

00 

9 

9 

.1 

48 

4-8 
9f. 

47 
4-7 

I 

9 

9-44253 
9.44297 

9-44341 
9.44385 
9.44428 

44 
44 
44 
43 
44 

9-45  987 
9-46035 
9.46082 
9.46  130 
9.46  177 

48 
47 
48 
47 

0.54013 
0.53965 
o.539i8 
0.53870 
0.53823 

9.98266 
9.98  262 
9.98259 

9.98255 
9.98251 

4 
3 

4 
4 

55 
54 
53 
52 
|l 

•3 
•4 

7 

•° 
14.4 
19.2 
24.0 
28.8 
33  6 

9-4 
14.1 
18.8 

III 

72   Q 

110 
ii 

12 
13 
14 

9-44472 
9  44  5l6 
9-44559 
9.44602 
9.44646 

44 
43 
43 
44 
43 

9.46224 
9.46271 
9-463I9 
9-46366 
9.46413 

47 
48 
47 
47 
47 

0.53776 
0.53729 
0.53681 

0.53634 
0.53587 

9.98248 
9.98244 
9.98240 
9.98237 
9-98233 

3 

4 
4 
3 

4 

60" 

% 
% 

8 
•9 

P-4 
43-2 

46 

37-6 
42.3 

45 

15 

16 

\l 

19 

9.44689 
9-44  733 
9-44  776 
9-448I9 
9.44862 

44 
43 
43 
43 

9.46460 
9.46  507 

9.46554 
9.46601 
9.46648 

47 
47 
47 
47 
<i6 

0-53540 
0-53493 
0.53446 
0-53399 
0.53352 

9.98  229 
9  .  98  226 

9.98  222 
9.98218 
9.98215 

3 

4 
4 
3 

45 
44 
43 
42 
41 

.1 

.2 

•3 
•4 

4.6 
9-2 

13.8 

18.4 
23.0 

4-5 
9.0 

'3-5 
18.0 
22.5 

20 

21 

22 

23 
24 

9-44905 
9.44948 
9.44992 
9-45035 
9-45  077 

43 
44 
43 
43 

9-46694 
9.46  741 
9.46  788 
9-46835 
9.46881 

47 
47 
47 
46 

0.53306 
0.53259 
0.53212 
0-53  165 
0-53  "9 

9.98  211 
9.98  207 
9.98204 
9.98  200 
9.98  196 

4 
4 
3 

4 
4 

40 

9 
9 

* 

•9 

2y.  6 
32.2 
36.8 
41.4 

27.0 

3J-5 
36.0 
40.5 

% 
% 

29 

9-45  "O 

9-45  163 
9-45  206 
9-45  249 
9-45  292 

43 
43 
43 
43 

42 

9.46  928 

9-46  975 
9.47021 
9.47068 
9-47  "4 

47 
46 
47 
46 

46 

0.53072 
0.53025 

0.52979 
0.52932 
0.52886 

9.98  192 
9.98  I§9 
9.98  IS? 

9.98  181 
9-98  177 

3 

4 
4 

4 

35 
34 
33 
32 
31 

.1 

.2 

•3 

44 

31 
I?1 

43 

3:1 

12.9 

172 

130 

31 

62 
33 
34 

9-45  334 
9-45  377 
9.45419 
9.45462 

9-45504 

43 
42 
43 
42 

9.47  160 
9.47207 

9.47253 
9.47299 
9-47346 

47 

46 

46 
47 
d.6 

0.52840 

0.52  793 
0.52  747 
0.52  701 
0.52654 

9-98  174 
9.98  170 
9.98  166 
9.98  162 
9-98  159 

4 
4 
4 
3 

30 

27 
26 

1 

.0 

I/.U 

22.0 
26.4 
30.8 

35-2 
39-6 

l/.^ 

21.  5 

25.8 
30.1 

9 

9 

39 

9-45  547 
9-45  589 
9-45632 
9-45674 
9-45  7i6 

42 
43 
42 
42 

9-47392 
9.47438 
9.47484 
9-47530 
9-47576 

46 
46 
46 
46 
46 

0.52608 
0.52562 
0.52  516 
0.52470 
0.52424 

9-98  155 
9.98  151 
9.98  147 
9.98  144 
9.98  140 

4 
4 
4 
3 

4 

25 
24 

23 

22 
21 

.1 

2 

4« 

4-2 

8  4 

41 

£ 

40 

4i 
42 
43 
44 

9-45  758 
9.45801 

9-45  843 
9-45885 
9-45927 

43 
42 
42 
42 

9.47622 
9.47668 

9-47  7H 
9.47  760 
9.47806 

46 
46 
46 
46 

A.6 

0.52378 
0.52332 
0.52  286 
0.52  240 
0.52  194 

9.98  136 
9.98  132 
9.98  129 

9-98  12? 
9.98  121 

4 
4 
3 

4 
4 

20 

JQ 

il 

•3 
•4 

i 

.7 

12.6 

16.8 

21.0 

25.2 

29.4 

"3 

16.4 

20.5 

24.6 

28.7 

45 
46 

47 
48 
49 

9  45969 
9.46011 
9-46053 
9.46095 
9.46  136 

42 
42 
42 
4i 

9.47852 
9.47897 

9-47943 
9.47989 
9-48035 

45 
46 
46 
46 

0.52  148 
0.52  103 
0.52057 
0.52011 
0.51965 

9.98  117 
9.98113 

9.98  1  10 
9.98  106 

9.98  IO2 

4 

4 
3 

4 
4 

'5 
H 
13 

12 
II 

.8 
•9 

33-6 
37-8 

4 

32.8  1 

36.9 

3 

50 

Si 

52 
53 

1  54 

9  46  178 
9.46  220 
9.46  262 
9-46303 
9-46345 

42 
42 
4i 
43 

9.48080 
9.48  126 
9.48  171 
9.48217 
9.48262 

46 
45 
46 
45 

0.51  920 
0.51874 
0.51  829 
0.51  783 
0.51  738 

9,98098 
9.98094 
9.98090 
9.98087 
9.98083 

4 

4 
4 
3 

4 

10 

I 
I 

.1 

.2 

•3 
•4 
•5 

0.4 
0.8 

1.2 

1.6 

2.0 

0.3 
0.0 
0.9 
1.2 

«-J 

9 

9 

59 

9.46386 
9  .  46  428 
9.46469 
9.46511 
9-46552 

42 
4i 
42 
4» 

9.48307 

9.48353 
9.48398 

9-48443 
9.48489 

46 

45 
45 
46 

0.51  693 
0.51  647 
0.51  602 
0.51557 
0.51  5ii 

9.98079 
9.98075 

9.98071 
9.98067 

9.98063 

4 

'   4 
4 
4 
4 

5 

4 
3 

2 
I 

.6 
•9 

li 
H 

1.8 

2.1 
2.4 
2.7 

60 

9.46594 

9.48534 

0.51  466 

9.98  060 

3 

0 

L.  Cos. 

d. 

L.  Cot?. 

c.d. 

L.  Tan?. 

L.  Sin. 

d. 

f 

r 

top. 

Pts. 

73° 

LOGARITHMS  OF  SINE,  COSINE,  TANGENT  AND  COTANGENT,  ETC.     47 


17°                                       1 

t 

L.  Sin. 

d. 

L.  Tang. 

c.d. 

L.  Cotg. 

L.  Cos. 

d. 

Prop.  Pts.     \ 

0 

2 

3 
4 

9-46594 
9-46635 
9.46  676 
9.46717 
9-46758 

41 
4* 
41 
41 
42 
4» 
4* 
41 
4< 
4» 
4° 
41 
41 
4< 
41 
40 
4> 
40 
4» 
40 

4> 
40 
4' 
40 
40 

41 
40 
40 
4<> 
40 

4° 
4<> 
40 
4<> 

40 

4<> 
40 
39 
40 
40 

39 
4<> 
40 
39 
40 

39 
40 
39 
39 
39 
40 
39 
39 
39 
39 
39 
39 
39 
39 
39 

9-48534 
9.48579 
9.48  624 
9.48669 
9.48  714 

45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
44 
45 
45 
44 
45 
44 
45 
44 
45 
44 
45 
44 
44 
45 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
43 
44 
44 
44 
43 
44 
43 
44 
43 
44 
43 
44 
43 
43 
44 
43 
43 
43 
43 
43 
44 
43 
43 

0.51  466 
0.51  421 
0.51  376 

0.51331 
0.51  286 

9  .  98  060 
9.98056 
9.98052 
9  .  98  048 
9.98044 

4 
4 
4 
4 
4 
4 
4 
3 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
3 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
5 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 

00 

f* 
y 

.1 

.2 

•  3 
•4 

:i 

.7 

.8 
•9 

.1 

.2 

•3 

•  4 

j 

•9 

I 

.2 

•  3 

•  4 

;2 

•9 
.1 

.2 

•3 

•  4 

.b 

:l 

•9 

.2 

•3 
•4 

j 

-9 

45 

4-5 
9.0 

!3i 

22.5 
27.0 

3J-S 
36.0 

40-5 

43 

tt 

12.9 
17.2 
21-5 
258 
30.1 
34-< 
38-7 

V 

41 

8^2 

12.3 

16.4 

20.5 

24.6 

28.7 

32.8 

36-9 

39 

?:l 

11.7 
15.6 
iQ-5 
23-4 
27-3 
31.2 

35-i 

4 
0.4 
0.8 

-1.2 

1.6 

2.0 

2-4 
2.8 

1-6 

44 

si 

o.o 

'3-2  , 
17  6 

22   0 
26.4 
30.8 

35-2 
39-6 

42 

4-2 
8.4 

12.6 

16.8 

21.  0 

25-2 
29.4 
336 

37-8 

t 
40 
4.0 
8.0 

12.0 

16.0 
20.  o 

24.0 
28.0 

32.0 

36.0 

5 

0.5 

1.0 

15 

2.0 
2-5 

30 

35 
4-0 
4-5 

3 

°-3  | 
0.6 

0.9 

1.2 

!:i 

2.1 
2.4 
2.7 

I 

I 

9 
10 
ii 

12 
13 

14 

9  .  46  800 
9.46841 
9.46882 
9.46923 
9.46964 

9-48  759 
9.48804 
9.48849 
9-48894 
9-48939 

0.51  241 
0.51  196 

0.51  151 
0.51  106 
0.51  061 

9  .  98  040 
9  .  98  036 
9.98032 
9.98029 
9.98025 

55 
54 
53 
52 
5i 

9.47005 

9-47045 
9.47086 
9-47  127 
9.47168 

9.48984 
9.49029 

9-49073 
9.49  118 
9-49  103 

0.51  016 
0.50971 
0.50927 

O.5O  8\>2 

0.50837 

9.98021 
9.98017 
9.98  013 
9  98  oc  i 
9  .  98  005 

50 

3 

8 

15 

16 

17 
18 

19 

9.47209 
9.47249 
9.47290 
9-47330 
9-47371 

9-49  207 
9-49252 
9.49296 
9-49341 
9.49385 

0.50793 
0.50  748 
o  .  50  704 
0.50659 
0.50  615 

9.98001 
9-97997 
9-97993 
9.97989 
9.97986 

45 
44 
43 
42 
41 

20 

21 
22 

23 
24 

9.47411 
9.47452 
9-47492 
9-47533 
9-47573 

9-49430 
9-49474 
9.49519 
9-49  563 
9.49607 

0.50570 
0.50526 
0.50481 

0.50437 
0.50393 

9.97982 
9.97978 

9-97974 
9.97970 
9.97966 

40 

3 

11 

II 
% 

29 

9-47613 
9.47654 
9.47694 
9  47  734 
9-47  774 

9.49652 
9.49696 
9-49  740 
9.49784 
9.49828 

0.50  348 
0.50304 
0.50  260 
0.50216 

o  .  50  1  72 

9.97962 
9-97958 
9-97954 
9-97950 
9.97946 

35 
34 
33 
32 
31 

mT 

?8 

11 

30 

3i 

32 
33 
34 

9.47814 
9-47854 
9.47894 

9-47934 
9  47974 

9.49872 
9.49916 
9.49960 
9  .  50  004 
9  .  50  048 

0.50  128 
o  .  50  084 
o  .  50  040 

0.49996 
0.49952 

9.97942 
9-97938 
9-97934 
9.97930 
9.97926 

9 

% 

39 

9.48014 
9-48054 
9.48094 
9  48  133 
9-48  173 

9.50092 
9.50  136 
9.50  180 
9.50223 
9.50267 

0.49  908 
0.49  864 
0.49820 

0.49777 

0-49  733 

9.97922 
9.97918 
9.97914 
9.97910 
9.97906 

25 
24 

23 

22 
21 

40 

4i 

42 

43 
44 

9.48213 
9-48252 
9.48292 
9-48332 
9-4837I 

9-503" 
9.50355 
9-50398 
9.50442 
9-50485 

0.49  689 
0.49645 
0.49  602 
0.49558 
OA951S 

9.97902 
9-97898 
9.97894 
9.97890 
9.97886 

20 

:§ 

\i 

45 
46 
47 
48 
49 

9.48411 
9.48450 
9.48490 

9-48529 
9.48568 

9-50529 
9.50572 
9.50616 
9-50659 
9-50703 

0.49471 
0.49428 
0.49  384 
0.49341 
0.49  297 

9.97882 
9.97878 
9.97874 
9.97870 
9.97866 

15 
14 

13 

12 
II 

~w 

I 

5 
4 
3 

2 
I 

~0" 

50 

5i 
52 
53 
54 

9.48  607 
9.48647 
9.48686 

9-48725 
9.48  764 

9.50746 

9-50789 
9-50833 
9.50876 
9.50919 

0.49  254 

0.49  211 
0.49  167 
0.49  124 
0.49081 

9.97861 
9-97857 
9-97853 
9-97849 
9-97845 

P 

57 
58 
59 

w 

9-48803 
9.48842 
9.48881 
9.48920 
9-48959 

9.50962 
9-5IOOS 
9-51048 
9-51092 
9-51  135 

0.49038 
0.48995 

o  48  952 

0.48908 
0.48865 

9.97841 
9-97837 

9-97833 
9.97829 
9-97825 

9-48998 

9-51  178 

0.48822 

9.97821 

L.  Cos. 

d. 

L.  Cotg. 

c.d. 

L.  Tang. 

L.  Sin. 

d. 

1 

Prop.  Pts. 

72° 

TABLE  IV. 


18° 

t 

L.  Sin. 

d. 

L.  Tang. 

c.d. 

L.  Cotg. 

L.  Cos. 

d. 

Prop.  Pts. 

Y 

I 

2 

3 
4 

9.48998 

9.49037 
9.49076 

9-49  H5 
9  49  153 

39 
39 
39 
38 
39 

39 
38 
39 
39 
33 

39 
38 
28 
39 
38 
38 
39 
38 
38 
38 
38 
38 
38 
38 
38 
38 
38 
38 
38 
38 

37 
38 
38 
37 
38 
38 
37 
38 
37 
37 
38 
37 
37 
38 

9-5i  178 

9.51  221 
9.51  264 
9.51  306 

9-51  349 

43 

43 
42 
43 
43 
43 
43 

43 
43 
42 
43 
43 
42 

43 
42 

42 

43 
42 

43 
42 
42 
42 
43 
42 

42 
42 
42 
42 
42 
42 
42 
42 
42 

42 
42 
42 
42 

42 

4>I 

42 

42 

41 
4* 
42 
41 
42 
41 

42 

4* 
41 
41 

0.48822 
0.48  779 
0.48  736 
0.48  694 
0.48  651 

9.97821 
9.97817 
9.97812 
9.97808 
9.97804 

4 
5 

4 
4 
4 
4 
4 
4 
4 
5 
4 
4 
4 
4 
4 
5 
4 
4 
4 

00 

59 
58 

JL 

55 
54 
53 
52 

.1 

.2 

•3 
•4 

^9 
.1 

.2 

•3 
•4 

•9 
.1 

.2 

•3 
•4 

'9 
.1 

.2 

•3 
•4 

ii 

•9 

4 
8 

12 

17 
21 

25 
30 

•3 
•4 

:i 

•9 

l 
2 
/ 
ii 

i< 

ic 

2< 
23 
3' 

3f 

: 

^ 
\ 

i 
ii 
\\ 

22 
2 
2( 
3; 

( 

t 

t 
i 

3 

i 

•9 

.2 

i 

•4 
•  7 

< 

\ 

12 

ie 

2C 
2^ 
2* 

I 

9 

•9 

3 

>-5 
•  4 
'•3 

.2 

-I 

17 

1   7 
r-4 
.1 

1:1 

1.2 

>-9 
)-6 

J-3 

5 
).f 

I.G 

z.o 

2-5 

J-o 
5-5 
l-° 
1-5 

43 

4-2 
8-4 

12.6 

16.8 

21.0 
25.2 
29.4 

33  6 
37-8 

x 

••I 

f  2 

•3 

'•4 
>.5 

5 

>_9 

38 

3-8 
7-6 
ii.  4 
15-2 

19.0 

22.8 
26.6 
30.4 
34-2 

36 
3.6 

10.8 
18  o 

21  6  ; 
25.2 
28.8 
32-4 

•4 

;  3-4  i 

0.8 

1.2 

1.6 

2.O 

2.4 

2.8 

I 
I 

9 

12 

14 

9.49  192 
9.49231 
9.49269 
9.49308 
9-49  347 

9.51392 
9.5I435 
9-51  478 
9-51  520 
9  51  563 

0.48608 
0.48  565 
0.48  522 
z  48480 
?  48  437 

9.97800 
9.97796 
9.97792 
9-97  788 
9.97784 

9.49424 
9.49462 
9.49500 
9  49539 

9.51  606 
9.51  648 
9.51691 

9-51  734 
9.51  776 

0.48394 
0.48352 
0.48309 
0.48  266 
0.48  224 

9-97779 
9-97775 
9.97771 

9-97767 
9-97  763 

50 

49 
48 

15 
16 

\l 

19 
"20" 

21 
22 
23 
24 

9-49577 
9.49615 

9  49654 
9-49692 
9-49730 

9.51819 
9.51  861 

9-51  903 
9.51946 

9-51988 

0.48  181 

0.48  139 
0.48097 
0.48  054 
0.48012 

9-97  759 
9-97754 
9-97750 
9-97746 
9-97  742 

45 
44 
43 
42 
41 

9.49768 
9.49  806 
9.49844 
9.49882 
9.49920 

9.52031 
9-52073 

9-52  157 
9.52  200 

0.47969 
0.47927 
0.47  885 

0-47843 
0.47  800 

9.97738 
9-97734 

9.97725 
9-97  72i 

4 
5 

4 
4 
4 
4 
5 
4 
4 
4 
5 
4 
4 
4 
5 
4 
4 
4 
5 
4 
4 
4 
5 
4 
4 
4 
5 
4 
4 
5 
4 
4 
5 
4 
4 
5 
4 
4 
5 
4 

40 

9 
9 

? 

29 

31 
32 

33 

f 
9 

39 

9  49958 
9.49996 

9.50034 
9.50072 
9.50  no 

9.52242 
9.52284 
9.52326 
9-52368 
9.52410 

0-47  758 
0.47716 
0.47674 
0.47632 
0.47590 

9.97717 
9-97713 
9-97  7oS 
9.97704 
9.97700 

35 
34 
33 
32 

W 

2Q 

28 

9.50  148 
9.50185 
9-50223 
9.50261 
9.50298 

9.52494 
9.52536 

9.52  620 

0.47548 

0.47  5°6 
0.47464 
0.47422 
0.47380 

9.97696 
9.97691 
9-97687 
9.97683 
9.97679 

9-50336 
9.50374 
9.50411 

9  50449 
9.50486 

9.52  661 
9-52  703 
9.52745 
9-52  787 
9.52829 

0-47  339 
0.47297 

0-47255 
0.47213 
0.47  171 

9.97670 
9.97666 
9.97662 
9-97657 

25 
24 
23 
22 
21 

40 

42 
43 

1  44 

9-50523 
9  50  561 
9.50598 
9  50  635 
9-50673 

9.52870 
9  52912 
9  52953 
9  52  995 
9-53037 

0.47  130 
o  47  088 
o  47  047 
o  47005 
0.46963 

9-97653 
9.97649 
9-97645 
9-97640 
9-97636 

20 

19 

17 
16 

45 
46 

9 

49 

9.50710 

9.50747 
9.50784 
9.50  821 
9.50858 

37 
37 
37- 
37 
38 

37 
37 
37 
36 

9-53078 
9-53  120 
9  53  161 
9.53202 

0.46  922 
0.46880 
0.46  839 
0.46  798 
0.46  756 

9-97632 
9.97628 
9-97623 
9.97619 
9-976I5 

IS 
14 
13 

12 
II 

50 

52 
53 
54 

9  .  50  896 

9.50933 

9.50970 
9.51  007 
9.51  043 

9-53285 
9.53327 
9.53368 
9-53409 
9.53450 

0.46  715 
0.46673 
0.46  632 
0.46591 
0.46  550 

9.97  610 
9.97606 
9.97602 
9  97597 
9-97593 

10 

I 

59 

9.51  080 
9  5i  "7 
9-51  154 
9.51  191 
9.51  227 

37 
37 
37 
36 

9-53492 
9-53533 
9-53574 
9-53615 
9-53656 

0.46  508 
0.46  467 
0.46  426 
0.46  385 
0.46344 

9-97589 
9-97584 
9.97580 
9-97576 
9-97571 

5 
4 
3 

2 
I 

GO 

9.51  264 

37 

9-53697 

0.46303 

9-97567 

0 

L.  Cos. 

d. 

L.  Cotg. 

c.d. 

L.  Tang. 

L.  Sin. 

d. 

t 

Prop.  Pts. 

!                                         71° 

LOGARITHMS  OF  SINE,  COSINE,  TANGENT  AND  COTANGENT,  ETC.     49 


19°                                    ! 

f 

L.  Sin. 

d. 

L.  Tang. 

c.d. 

L.  Cotg. 

L.  Cos. 

d. 

Prop.  Pts.     | 

0 

I 

2 

3 
4 

9.51  264 
9-5i3oi 
9-5I338 
9-51  374 
9-51  4" 

37 
37 
36 
37 
36 

37 
36 
37 
36 
36 

37 
36 
36 
36 
37 
36 
36 
36 
36 
36 
36 
36 
36 
36 
36 

36 
35 
36 
36 
36 

35 
36 
35 
36 
35 
36 
35 
36 
35 
36 
35 
35 
36 
35 
35 
35 
35 
35 
35 
35 
36 
34 
35 
35 
35 
35 
35 
35 
34 
35 

9-53697 
9-53  738 
9-53779 
9-53820 
9-5386I 

41 
41 
41 
41 
4» 
4< 
4< 

4* 
40 
4i 

4» 
40 
41 
41 
4<> 
4» 
4<> 
4^ 
40 
41 

0.46303 
0.46  262 

0.46  221 

0.46  180 

0.46  139 

9-97567 
9-97563 
9.97558 
9-97554 
9-97550 

4 
5 
4 
4 
5 
4 
5 
4 
4 
5 
4 
4 
5 
4 
5 
4 
5 
4 
4 
5 
4 
5 
4 
5 
4 
4 
5 
4 
5 
4 
5 
4 
5 
4 
5 
4 
5 
4 
5 
4 
5 
4 
5 
4 
5 
4 
5 
5 
4 
5 
4 
5 
4 
5 
4 
5 
5 
4 
5 
4 

(JO 

59 
58 

H 

.1 

.2 

.3 
.4 

i 

•9 
.1 

.2 

•3 
•  4 

:S 
:i 

•9 
.1 

.2 

•3 
•4 

:i 

:l 

•9 
.1 

.2 

•3 
•4 

:1 

•9 

41 

4-1 

8.2 

"•3 

16.4 
20.5 
24.6 
28.7 
32.8 

36.9 
3 

.'      3 

.2     7 

•3    " 
•4    15 

•5    19 
.6    23 
.7    27 
•8    3i 
•9   35 

37 
3-7 
7-4 
ii.  i 

14.8 
18.5 

22.2 

25.9 
29.6 

33-3 
35 

3-5 
7-o 
10.5 
14.0 
17-5 

21.0 

24-5 
28.0 

31.5 

5 

o-5 

I.O 

15 

2.0 
2-5 

3-o 

3-5 
4.0 

4-5 

40 

4.0 
8.0 

12.  0 

16.0 

20.0 
24.0 
28.0 
32.0 
36.0 

9 

1 

iiii 

-5 
•  4 
•3 

.2 
.1 

36 

3-6 

10.8 
14.4 
18.0 

21.6 

25.2 
28.8 
32.4 

34 

i:48 

10.2 
136 
17.0 
20.4 
23-8 
27.2 
30.6 

4 
0.4 
0.8 

1.2 

1.6 

2.0 

:i 

i:l 

I 
I 

9 

'To 
ii 

:2 
13 

14 

9-51  447 
9.51  484 

9-51  52o 
9.51557 
9-5i  593 
9-51629 
9.51  666 
9.51  702 
9  5'  738 
9  5i  774 

9-53902 
9-53943 
9-53984 
9-54025 
9.54065 

0.46  098 
0.46057 

o  46016 
0-4597 
0.45  93 

9-97545 
9-97541 
9-97536 
9-97532 
9.97528 

55 
54 
53 
52 
5i 

9.54106 
9-54147 
9-54  187 
9.54228 
9.54269 

0.45894 
0-45  853 
0.45813 
0.45  772 
0-45  731 

9-97523 
9  975*9 
9-975I5 
9.97510 
9.97506 

50 

49 
48 

47 
46 

IS 

\l 

ft 

21 
22 
23 

24 

9.51  811 

9-5I847 
9.51883 

9-5i  9i9 
9-5I955 

9-54309 
9-54350 
9  54390 
9  54431 
9-54471 

0.45  691 
0.45  650 
0.45  610 
0-45  569 
0-45  529 

9.97501 

9-97497 
9.97492 
9.97488 
9.97484 

45 
44 
43 
42 
41 
40~ 
39 
38 

P 

9  51  99i 
9.52027 
9.52063 
9.52099 
9-52  135 

9  54512 
9  54552 
9  54593 
9  54633 
9  54673 

40 
41 
40 
40 
4» 

40 
40 
41 
40 
4«> 
40 
4<> 
40 
4° 
4<> 
40 
4<> 
40 
4° 
4° 
4<> 
4<> 
39 
4° 
4«> 
4° 
39 
40 
4<> 
39 
40 
39 
40 
39 

o  .  45  488 
0.45  448 
0.45407 
0-45  367 
0.45327 

9-97479 
9-97475 
9.97470 
9.97466 
9.97461 

111 

3 

29 

9.52  171 
9.52207 
9.52242 
9.52278 
9  52  3H 

9  547H 
9-54754 
9-54794 
9.54835 
9  54875 

0.45  286 
0.45  246 
0.45  206 
0.45  165 
0.45  125 

9  97457 
9  97453 
9.97448 

9-97444 
9-97439 

35 
34 
33 
32 
3i 

80 

3i 
32 
33 
_34_ 

9 

3 

39 

9-52350 
9-52385 
9.52421 

9-52456 
9-52492 

9-54915 
9-54955 
9-54995 
9-55035 
9.55075 

0.45  08 
0.4504 
0.4500; 
0.4496; 
0.4492 

9  97435 
9  9743° 
9.97426 

9-97421 
9.97417 

30 

29 
28 

3 

9-52527 
9-52563 
9-52598 
9-52634 
9.52669 

9-55  "5 
9-55  155 
9-55  195 
9.55235 
9  55275 

0.44885 
0.44845 
0.44805 
0.44  765 
0-44  725 

9  97412 
9.97408 
9-97403 
9  97399 
9  97394 

25 
24 
23 

22 
21 

20" 

19 

18 

!| 

40 

4i 

42 

43 

44 

9  52  705 
9.52740 

9.52775 
9.52  811 
9.52  846 

9  553J5 
9-55355 
9-55395 
9-55434 
9-55474 

0.44685 
0.44645 
0.44605 
0.44566 
0.44526 

9.97390 

9  97385 
9.97381 
9.97376 
9.97372 

11 
% 

49 

Iso' 

1  51 
52 
53 
54 

9.52881 
,9.52916 
9.52951 
9-52986 
9  53021 

9-555H 
9-55554 
9-55593 
9.55633 

o  .  44  486 
0.44446 
0.44407 
0.44367 
0.44327 

9  97367 
9  97363 
9-97358 
9  97353 
9  97349 

15 
H 
13 

12 
II 

9-53056 
9-53092 
9-53  126 
9-53i6i 
9-53I96 

9-55712 
9  55752 
9-55  79i 
9-55831 
9  55870 

0.44288 
0.44248 
0.44209 
0.44  169 
0.44  130 

9-97344 
9-97340 
9-97335 
9-97331 
9.97326 

10 

6 

P 
P 

59 

9-53231 
9-53266 

9-53301 
9.53336 
9-53370 

9-55910 
9-55949 
9.55989 
9.56028 
9.56067 

39 
4<> 
39 
39 
4<> 

0.44090 
0.44051 
0.44011 
0.43972 
0-43933 

9-97322 

9-973I7 
9.97312 
9.97308 
9  97303 

5 
4 
3 

2 
I 

60 

9  534^5 

9-56107 

0.43893 

9.97299 

0 

L.  Cos. 

(1. 

L.  Cots. 

c.d. 

L.  Tang. 

L.  Sin. 

d. 

'  f 

Prop.  Pts. 

70° 

TABLE  IV. 


20° 

0 

L.  Sin. 

d. 

L.  Tang. 

c.d. 

L.  Cotg. 

L.  Cos. 

d. 

p 

rop. 

Pts. 

0 

I 

2 

3 
4 

9.53405 
9-53440 
9-53475 
9.53509 
9-53544 

35 
35 
34 
35 
34 

9.56  107 
9.56  146 
9-56185 
9.56224 
9  .  56  264 

39 
39 
39 
40 

39 

0.43893 
0.43  854 

0.43  815 
0.43  776 
0.43  736 

9.97299 

9-97294 
9.97  289 
9.97285 
9.97280 

5 

5 
4 
5 

00 

59 
58 

i 

.1 

2 

40 

\l 

39 

3-i 

7   £ 

I 
I 

9 

9.53578 
9.53$i3 
9.53647 
9-53682 

9-53  7i6 

35 
34 
35 
34 
35 

9-56303 
9.56342 
9-56381 
9.56420 

9.56459 

39 
39 
39 
39 
39 

0.43  697 
0.43658 
0.43619 
0.43  580 
0.43  54i 

9.97276 
9.97271 
9.97266 
9.97262 
9-97257 

5 
5 

4 
5 

55 
54 
53 
52 
Si 

•3 

:! 

.7 

12.0 

16.0 

20.0 

7.0 

ii.  7 
15.6 
19-5 
23  4 
27.7 

10 

ii 

12 

13 
14 

9-53  751 
9-53  7«5 
9-53819 
9.53854 
9.53888 

34 
34 

35 
34 

34 

9.56498 
9.56537 
9.56576 
9.56615 
9-56654 

39 
39 
39 
39 
39 

0.43  502 

0.43463 
0.43  424 

0-43  385 
0-43  346 

9-97252 
9.97248 

9-97243 
9-97238 
9-97234 

4 
5 
5 

4 

w 

8 

% 

.8 
•9 

32.0 
36.0 

38 

31-2 
35-1 

37 

15 

16 

!i 

19 

9-53922 
9-53957 
9-53991 
9-54025 

9  54059 

35 
34 
34 
34 
34 

9.56693 
9.56732 
9.56771 
9.56810 
9.56849 

39 
39 
39 
39 
38 

0-43  307 
0.43  268 
0.43  229 
0.43  190 
0.43  I5i 

9.97229 
9-97224 
9.97220 
9-97215 
9.97210 

5 
4 
5 
5 

45 
44 
43 
42 
41 

.2 

•  3 
•4 

3^ 

7-6 
11.4 
15-2 
19.0 

3-7 
7-4 
ii.  i 
14.8 
18.5 

20 

21 

22 
23 

24 

9.54093 
9-54I27 

9.54i6i 

9-54I95 
9-54229 

34 
34 
34 
34 
34 

9-56887 
9.56926 

9-56965 
9.57004 
9.57042 

39 
39 
39 
38 
39 

0-43  "3 
0.43074 

0.43035 
0.42  996 
0.42958 

9.97206 
9.97201 

9-97  196 
9.97192 
9-97  187 

5 

5 

4 
5 

40 

3 

11 

•i 

•9 

22.8 
26.6 
30.4 

34-2 

22.2 

25-9 
29.6 

33-3 
> 

3 

2 

29 

9-54263 
9-54297 
9-54331 
9-5436? 
9-54399 

34 
34 
34 
34 
34 

9.57081 

9.57  120 
9.57158 
9-57197 
9.57235 

39 

38 
39 
38 
39 

0.42919 
0.42880 
0.42  842 
0.42  803 
(^42  765 

9.97  182 
9.97178 

9-97  173 
9.97  168 

9-97  163 

4 
5 
5 
5 

35 
34 
33 
32 
3i 

.1 

.2 

•3    i< 

A          \. 

35 
J-S 

7-0 
>-5 

1  O 

80 

3i 
32 
33 
34 

9-54433 
9.54466 

9-54500 
9-54534 
9.54567 

33 
34 
34 
33 

OJ 

9.57274 
9.57312 
9-57351 
9.57389 
9.57428 

38 
39 
38 
39 
08 

0.42  726 
0.42688 
0.42  649 
0.42  611 

0.42  572 

9-97  159 
9-97  154 
9-97  149 
9-97  H5 
9-97  HO 

5 
5 

4 
5 

30 

1 

•5    * 

.6     2 

.7     2, 

.8    2! 
•9   3 

7-3 

I.O 

ti 

[.5 

9 

!? 

39 

9-54601 
9-54635 
9.54668 
9.54702 
9-54735 

34 
33 
34 
33 
34 

9.57466 
9.57504 
9-57543 
9.57581 
9-57619 

38 
39 
38 
38 

30 

0.42534 
0.42496 

0.42457 
0.42419 
0.42  381 

9  97135 
9-97  130 
9.97  126 
9.97121 
9.97  116 

5 
5 

4 
5 
5 

25 
24 
23 

22 
21 

.1 

2 

34 

n 

33 
36'36 

40 

4i 

42 
43 
44 

9-54769 
9.54802 
9-54836 
9-54869 
9-54903 

33 
34 

33 
34 
33 

9.57658 
9.57696 

9-57734 
9-57772 
9.57810 

38 
38 
38 
38 

30 

0.42342 
0.42304 
0.42  266 
0.42  228 
0.42  190 

9.97  in 
9-97  107 
9.97  102 
9.97097 
9.97092 

5 

4 
5 
5 
5 

20 

J9 

II 

•3 
•4 

.7 

10.2 
13-6 
17.0 
20.4 
23.8 

9-9 
13.2 

i6.g 
19.8 

23-1 

s 

47 
48 

49 

9-54936 
9.54969 
9-55003 
9-55036 
9-55069 

33 
34 
33 
33 

9.57849 
9  57887 
9.57925 
9-57963 
9.58  ooi 

38 
38 
38 
38 
^s 

0.42  151 
0.42  113 
0.42075 
0.42037 
0.41  999 

9.97087 
9-97083 
9.97078 
9.97073 
9.97068 

5 
4 
5 
5 
5 

15 
14 
13 

12 
II 

.8 
•9 

27.2 
3O.6 

5 

26.4 
29.7 

4 

50 

5i 

52 
53 
54 

9-55  102 
9-55  136 
9-55  169 
9-55202 
9.55235 

34 
33 
33 
33 
q-a 

9-58039 
9.58077 
9-58  n5 
9.58153 
9-58  191 

38 
38 
38 
38 
38 

0.41  961 

0.41  923 
0.41  885 
0.41  847 
0.41  809 

9.97063 
9-97059 
9-97054 
9.97049 

9-97044 

5 
4 
5 
5 

5 

10 

§ 
I 

.1 

.2 

•  3 
•4 
•5 

o-5 

I.O 

i-5 

2.0 
2-5 

0.4 
0.8 
1.2 

1.6 

2.0 

55 
56 

ii 

59 

9-55268 
9-55301 
9-55334 
9.55367 
9-55400 

33 
33 
33 
33 

9-58229 
9-58267 
9-58304 
9-58342 
9-58380 

38 

37 
38 
38 

,Q 

0.41  771 

0.41  733 
0.41  696 
0.41  658 
0.41  620 

9-97039 
9.97035 
9.97030 
9.97025 
9.97020 

5 
4 
5 
5 
5 

5 
4 
3 

2 
I 

.b 

:l 

•9 

3-o 
3-5 
4-0 
4-5 

111 

5.8 

60 

9-55433 

9.58418 

0.41  582 

9.97015 

5 

0 

L.  Cos. 

d. 

L.  Cotg. 

c.d. 

L.  Tang. 

L.  Sin. 

d. 

t 

F 

rop.  ] 

Pte. 

69° 

LOGARITHMS  OF  SINE,  COSINE,  TANGENT  AND  COTANGENT,  ETC.     5 , 


21° 

r 

L.  Sill. 

d. 

L.  Tang. 

c.d. 

L.  Cotg. 

L.  Cos. 

d. 

Prop.Pte. 

0 

I 

2 

3 
4 

9-55433 
9.55466 

9-55499 
9-55  532 
9.55564 

33 
33 
33 
33 
33 
33 
33 
33 

33 

33 
33 
33 
33 
33 
33 

33 
33 
33 
33 
33 

33 
33 
33 
33 
33 
33 
33 
33 
33 
33 
3* 
3» 
33 
33 
3« 

3» 
33 

33 
33 
33 

33 
3' 
33 
33 
33 

3' 
33 
3* 
33 
33 

3* 
33 
3' 
3i 
33 

3» 
33 
3* 
3» 
33 

9-58418 
9-58455 
9.58493 
9-58531 
9-58569 

37 

38 
38 
38 
37 
38 
37 
38 
38 
37 
38 
37 
38 
37 
37 
38 
37 
38 
37 
37 
37 
38 
37 
37 
37 
37 
38 
37 
37 
37 
37 
37 
37 
37 
37 
37 
37 
36 
37 
37 
37 
37 
36 
37 
37 
37 
36 
37 
37 
36 

37 
36 
37 
36 
37 
36 
37 
36 
37 
36 

0.41  582 

0.41  545 
0.41  507 
0.41  469 
0.41  431 

9.97015 
9.97010 
9.97005 
9.97001 
9.96996 

5 

5 
4 
5 
5 
5 
5 
5 
5 
5 
4 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
4 

5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
6 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 

60 

P 
g 

.1 

.2 

•3 
•4 

:i 

i 

•9 
.1 

.2 
•3 

:! 

is 

•9 
.1 

.2 

•3 
•4 

:i 

•9 
.1 

.2 

•3 
•4 

i 

•9 

38 

3-8 
7-6 
11.4 
15-2 
19.0 

22.8 
26.6 
304 
34.2 

36 

3-6 

10.8 

14.4 
18.0 

21.6 

25.2 
28.8 
32.4 

J" 

.2        < 

•3     < 
.4    i: 

5    I( 

.6     K 

•7    * 
.8    l 
.9   2\ 

3> 

6.2 

93 
12.4 

21.7 
24.8 
27-9 

5 

°-5 

I.O 

1-5 

2.0 

2.5 

3-o 
3-5 
4.0 

4-5 

37 
3-7 
74 
II.  1 
14.8 
18.5 
22.2 
25.9 
29.6 

33-3 

33 

i:i 

99 
15.2 
i&.J 
19.8 
23.1 
26.4 
29.7 

J» 

J-a 

>-4 
)-6 

2.8 

3.0 

?-2 

|1 

3.8 

| 
o'.6 

1.2 

I  8 
2.4 
3-o 
36 

4-2 

48 

54 

4 
0.4  j 
0.8 

1.2 

1.6 

2.O 

;i 

3:8 

\ 

9 

\w 

II 

12 

13 

H 

9-55  597 
9-55630 
9-55663 
9-55695 
9.55728 

9.58606 
9.58644 
9.58681 
9-58719 
9-58757 

0.41  394 
0.41  356 
0.41  319 
0.41  281 
0.41  243 

9.96991 
9.96986 
9.96981 
9-96976 
9.96971 

55 
54 
53 
52 
5i 

9-55  76i 
9-55  793 
9.55  826 
9.55858 
9-5589I 

9.58794 
9-58832 
9-58869 
9-58907 
9-58944 

0.41  206 
0.41  168 
0.41  131 
0.41  093 
0.41  056 

9.96966 
9.96962 
9-96957 
9-96952 
9.96947 

60 

8 
8 

3 

:; 

19 

9.55923 
9^5956 

9.56021 
9-56053 

9.58981 
9.59019 
9-59056 
9-59094 
9-59131 

0.41  019 
0.40981 
0.40  944 
0.40  906 
0.40  869 

9.96942 
9.96937 
9.96932 
9.96927 
9.96922 

45 
44 
43 
42 
4i 

IT 

1 

20 

21 
22 
23 
24 

9.56085 
9.56  na 
9-56  150 
9.56  182 
9-56215 

9-59  1  68 
9-59205 

9-59243 
9.59280 

9.59317 

0.40832 
0.40  795 
0.40  757 
0.40  720 
0.40  683 

9.96917 
9.96912 
9.96907 
9  .  96  903 
9.96898 

§ 
2 

29 

9.56247 
9.56279 
9-56311 
9-56343 
9.56375 

9-59354 
9.59391 
9.59429 
9.59466 

9-59503 

0.40  646 
0.40609 
0.40571 
0.40  534 
0.40497 

9  .  96  893 
9.96888 
9-96883 
9.96878 
9.96873 

35 
34 
33 
32 
3i 

30 

3i 
32 
33 

34 

9.56408 
9.56440 
9.56472 
9-56504 
9-56536 

9-59540 
9-59577 
9-59614 
9-5965I 
9.59688 

9.59725 
9.59762 

9  59799 
9.59835 
9.59872 

0.40  460 

0.40423 
0.40386 
0.40349 
0.40312 

9.96868 
9-96863 
9-96858 
9-96853 
9.96848 

30 

11 
11 

25 
24 
23 

22 
21 

20- 

19 

18 

g 

35 
36 

9 

39 

9.56568 

9-56599 
9-56631 
9.56663 
9.56695 

0.40275 
0.40  238 

0.40  2OI 
O.40  165 
O.40  125 

9  96  843 
9-96838 
9-96833 
9.96828 
9-96823 

j40 

4i 
42 

43 
44 

9-56727 
9-56759 
9-56790 
9  56  822 
9.56854 

9-59909 
9^9946 
9.59983 
9.60019 
9.60  056 

0.40091 
0.40054 
0.40017 
0.39981 
0-39944 

9.96818 
9-96813 
9.96808 
9.96803 
9.96  798 

45 
46 

47 
48 
49 
50 
5i 
52 
53 
54 

9.56886 
9.56917 

9  56949 
9  .  56  980 
9.57012 

9.60093 
9.60  130 
9.60  1  66 
9.60203 
9  .  60  240 

0.39907 
0.39870 

o  39834 

0-39  797 
0.39760 

9-96  793 
9.96  788 
9-96  783 
9.96  778 
9.96772 

15 
H 
13 

12 
II 

"10" 

6 

9  57044 
9-57075 
9.57107 

9.57138 
9.57169 

9.60276 
9.60313 
9.60349 
9.60386 
9.60422 

0.39  724 
0.39687 
0.39651 
0.39614 
0-39  578 

9.96767 
9.96  762 

9.96757 
9.96  752 
9.96  747 

55 
56 

? 

ft 

9.57201 
9-57232 
9.57264 

9.57295 
9-57326 

9.60459 
9.60495 
9.60532 
9.60  568 
9.60605 

o  39  54i 
0.39505 
0.39468 

0.39432 
0-39395 

9.96742 
9.96  737 
9.96  732 
9.96727 
9.96  722 

5 
4 
3 

2 
I 

9-57358 

9.60641 

0-39359 

9.96717 

0 

L.  Cos. 

d. 

L.  Cots. 

o.  d. 

L.  Tang. 

L.  Sin. 

d. 

/ 

Prop.  Pts. 

68° 

TABLE  IV. 


22° 

I 

L.  Sin. 

d. 

L.  Tang. 

c.d. 

L.  Cotg. 

L.  Cos. 

d. 

Prop.  Pts. 

0 

I 

9.57358 
9-57389 

31 

9.60641 
9.60677 

36 

0-39359 
0.39323 

9.96717 
9.96711 

6 

60 

«>9 

2 

9.57420 

9.60714 

,6 

0.39  286 

9.96  706 

5 

S8 

37  1 

36 

3 

4 

9  57451 
9.57482 

32 

9.60750 
9.60  786 

36 
37 

0.39250 
0.39214 

9.96  701 
9.96696 

5 

5 

.1 

2 

3-7 

7  4 

3-6 

7.2 

i 

9.57514 
9-5754? 

9.60823 
9  .  60  859 

36 
,6 

0.39177 
o-39  HI 

9.96691 
9.96686 

5 

55 
S4 

•3 
.4 

II.  I 

14.8 

14.4 

I 

9 

9-57576 
9-57607 
9-57638 

3» 
3* 

9.60895 
9.60931 
9-60967 

36 
36 
37 

0.39  105 
0.39069 
0.39033 

9.96681 
9.96676 
9.96670 

5 
5 

6 

53 
52 

1 

18-5 
22.2 
25.9 

18.0 

21.6 

25-2 

10 

9.57669 

9.61  004 

,6 

0.38996 

9.96665 

50 

.8 

29.6 

28.8 

ii 

9.57700 

9.61  040 

og 

0.38960 

9.96  660 

5 

49 

•9 

33-3 

32-4 

12 
13 
14 

9-57731 
9.57762 
9-57793 

$ 

9.61  076 

9.61  112 
9.6l   148 

36 
36 

36 

0.38924 
0.38888 
0.38852 

9-96655 
9  .  96  650 
9.96645 

5 

5 

48 

8 

35 

\l 

9.57824 
9.57855 
9.57885 

30 

9.6l  184 
9.61  22O 
9.61  256 

36 
36 

,A 

0.38816 
0.38  780 
0.38  744 

9  .  96  640 
9.96634 
9  .  96  629 

6 
5 

45 
44 
43 

.2       7-0 

•3    lo-S 

18 

9.57916 

* 

9.61  292 

,jt 

0.38  708 

9.96624 

5 

42 

.4       I4.O 

19 

9-57947 

31 

9.61  328 

3° 
36 

0.38672 

9.96619 

5 

41 

20 

21 

9.57978 
9.58008 

30 

9-6l  364 
9.6l  400 

36 

0.38636 
0.38600 

9.96614 
9.96608 

6 

*? 

.7    24.5 

X     28  O 

22 
23 

24 

9-58039 
9.58070 
9.58  101 

9.6l  436 
9.6l  472 
9.6l  508 

3° 
36 
36 

36 

0.38  564 
0.38528 
0.38492 

9.96603 
9.96598 
9-96593 

5 
5 
5 

11 

9    3'-5 

11 

9-58131 
9.58  162 

9.61  544 
9.61  579 

35 
-e. 

0.38456 
0.38421 

9.96588 
9.96582 

6 

35 
34 

3» 

31 

•3     | 

% 

29 

9.58192 
9.58223 
9-58253 

30 

30 
31 

9.61  615 
9.61  651 
9.61  687 

3° 
36 
36 
35 

0-38385 
0.38349 
0.38313 

9-96577 
9.96572 
9.96567 

5 
5 
5 

33 
32 
31 

.2 
•  3 

A 

T?  8 

6.2 

9.3 

12.4 

30 

9  •  S8  284 

9.61  722 

ofi 

0.38278 

9  .  96  562 

30 

16  o 

31 

9-583I4 

3° 

9.61  758 

30 

,6 

0.38242 

9  96556 

29 

.6 

19.2 

18.6 

§ 

34, 

9-58345 
9.58406 

30 

9.61  794 
9.61  830 
9,61  865 

36 

35 

16 

0.38  206 
0.38  170 
0.38135 

9  96551 
9  96546 
9.96541 

5 
5 

6 

11 

•9 

22.4 
25.6 
28.8 

21.7 

24.8 

27.9 

9-58436 
9.58467 

3' 

9.61  901 
9.61  936 

35 

0.38099 
0.38064 

9  96535 
9  96530 

5 

25 

24 

39 

9-58497 
9-58527 
9  58557 

30 
30 

30 

9.61972 
9.62  008 
9.62043 

36 

36 
35 

,6 

0.38028 
0.37992 
0-37957 

9-9652? 
9.96520 
9.96514 

5 

5 
6 

23 

22 
21 

.1 

2 

30 

ft 

29 
2.9 

(0 

9-58588 

3 

9  .  62  079 

0.37921 

9.96509 

20 

.3 

9  ° 

8  7 

41 

9.58618 

3° 

9.62  114 

35 

0.37886 

9-96504 

5 

19 

•4 

12.0 

ii.  6 

42 

9.58648 

3° 

9.62  150 

30 

0.37850 

9.96498 

18 

15   0 

14-5 

43 

9.58678 

3° 

9.62  185 

35 
-f. 

0-37815 

9.96493 

5 

17 

.6 

18.0 

17-4 

44 

9.58  709 

31 

9.62  221 

3° 

0.37  779 

9.9648$ 

5 

ib 

.7 

21.0 

20.3 

| 

9-58739 
9.58769 

30 

9.62  256 

9  .  62  292 

36 

0-37744 
0.37708 

9.96483 
9.96477 

6 

15 
14 

•9 

24.0 
27.0 

23.2 
26.1 

9.58799 
9  -  58  829 

30 
30 

9.62327 
9.62  362 

35 
35 

0.37673 
0.37638 

9.96472 
9.96467 

5 
5 

13 

12 

49 

9-58859 

3° 

9.62  398 

0.37602 

9.96461 

II 

6 

5 

50 

9.58889 
9.58919 

30 

9.62433 
9.62468 

35 

0.37567 
0.37532 

9.96451 

5 

10 

.1 

.2 

0.6 

1.2 

0-5 

I.O 

52 

9-58949 

3° 

9.62  504 

36 

0.37496 

9-96445 

6 

8 

•3 

1.8 

15 

53 

9-58979 

30 

9.62  539 

35 

0.37461 

9.96440 

5 

7 

•4 

2.4 

2  0 

54 

9.59009 

30 

9.62574 

35 

0.37426 

5 
5 

6 

3-o 

2-5 

55 

9.59039 

9.62609 

0.37391 

9.96429 

5 

•  b 

3-6 

3-0 

56 

9.59069 

30 

9  .  62  645 

36 

0-37355 

9.96424 

5 

4 

'I 

4-2 

3-5 

57 

9.59098 

29 

9.62680 

35 

0.37320 

9.96419 

5 

3 

.8 

4-8 

4.0 

58 
59 

9-59  128 
9-59  IS8 

3° 
30 

9-62  715 
9.62  750 

35 
35 

0.37285 
0.37250 

9-96413 
9.96408 

6 
5 

2 
I 

•9 

5-4 

4-5 

60 

9.59188 

9.62  785 

0.37215 

9.96403 

5 

0 

L.  Cos. 

d. 

L.  Cotg. 

c.d. 

L.  Tang. 

L.  Sin. 

d. 

t 

Prop.  Pts. 

67°                                        | 

LOGARITHMS  OF  SINE,  COSINE,  TAff     -tfT  AND  COTANGENT,  ETC. 


i                                         23° 

t 

L.  Sin. 

d. 

L.  Tang. 

c.d. 

L.  Cotg. 

L.  Cos. 

d. 

Prop.  Pts. 

0 

I 

2 

3 
4 

9.59  188 
9.59218 
9-59247 
9-59277 
9-  593°7 

30 
29 
30 
30 
29 

30 
30 
29 
30 
29 

30 
29 
30 
29 
30 
29 
39 
30 
29 

«9 
30 
29 
29 
29 
*9 
30 

«9 

29 
29 
29 
29 
29 
29 
29 
«9 
29 
29 
29 
29 
28 
29 
29 
29 
28 
29 
29 
29 
28 
29 
28 
29 
» 

38 

29 
28 
39 
28 
29 
28 
28 

9-62785 
9  .  62  820 
9.62  855 
9  .  62  890 
9.62  926 

35 
35 
35 
36 
35 
35 
35 
35 
35 
34 
35 
35 
35 
35 
35 
35 
34 
35 
35 
35 
35 
34 
35 
35 
34 
35 
34 
35 
35 
34 
35 
34 
35 
34 
35 
34 
35 
34 
34 
35 
34 
34 
35 
34 
34 
35 
34 
34 
34 
34 
35 
34 
34 
34 
34 
34 
34 
34 
34 
34 

0.37215 
0.37  180 

0-37  145 
0.37  no 
0.37074 

9.96403 
9.96397 
9-96392 
9.96387 
9.96381 

6 
5 
5 
6 
5 
6 
5 
5 
6 
5 
6 
5 
5 
6 
5 
6 
5 
6 
5 
6 

5 
5 

6 
5 

6 

5 

6 
5 
6 
5 

6 
5 
6 
5 
6 

5 

6 
5 

6 
5 

6 
5 
6 
6 
5 
6 
5 
6 
5 
6 

6 
5 
6 

5 

6 

6 
5 
6 
5 
6 

00 

CQ 

i 

.1 

4 
•  3 
•4 

i 

9 
.1 

.2 
•3 

:l 
i 

•9 
.1 

.2 

•3 
-4 

i 
i 

•9 

3 

3 
7 

ID 

11 

21 

25 
28 
32 

.1 

.2 
•3 
•4 

:l 
.1 

•9 

2 

1 
s 

12 
IE 

ll 

21 
2<: 
2; 

.1 

.2 

•3 
.4 

i 

:l 

.9 

( 

: 

i 
i 

i 

6 

.6 

2 

.8 
•4- 

0 

.6 

.2 

.8 
•4 

I 

\ 

1C 

': 
i; 

2C 
2; 
2; 

3< 

0 

.0 
.0 

>.o 

.0 

«:S 

.0 

^o 

r.o 

i 
i 

i 

2 
2 

6 

>.6 

[.2 

[.8 
i-4 

M 
1:5 

;-4 

35 

3-5 

7.0  1 
10.5 
14.0 
17-5 

21.0 

24-5 
28.0 

31-5 

14 

S;i 

>.2 

t-6 

r.O 

;i 

r.2 

>.6 

39 

2.9 

ll 

II.  6 

14-5 
17-4 
20.3 

2:? 

•8 

2.8 

I'6 

1.4 

1.2 

4..0 
5.8 
9.6 
2.4 
52 

5 

0.5 

1.0 

'•5 

20 

2-5 
30 

35 
40 

4-5 

i 

I 

9 

lo- 
ii 

12 
13 

H 

9.59336 
9-59366 
9-59396 
9-59425 
9-59455 

9.62  961 
9.62  996 

9-63031 
9  .  63  066 
9.63  101 

0.37039 
0.37004 
0.36969 
0.36934 
o  .  36  899 

9-96376 
9.96370 

9-96365 
9-96360 
9-96354 

55 
54 
53 
52 
5i 

9-59484 
9-595H 
9-59543 
9  59573 
9.59602 

9-63  135 
9-63  170 
9.63205 
9.63240 
9  63275 

0.36865 

0.36830 

0.36  795 
0.36  760 
0.36  725 

9-96349 
996343 
9  96338 
9  96333 
9.96327 

60 

* 
% 

15 
10 

11 

19 

9-59632 
9  59661 
9.59690 
9.59720 
9-59749 

9.63310 
9.63345 
9.63379 
9.63414 

9-63449 

0.36690 
0.36655 
0.36621 
0.36586 
0.36551 

9-96322 
9.96316 
9.96311 

9-96305 
9.96300 

45 
44 
43 
42 

4i 
40" 

39 
38 

8 

20 

21 
22 

23 

24 

9-59778 
9-59808 

9  59837 
9.59866 

9  59895 

9.63484 
9.63519 
9.63  553 
9-63588 
9-63623 

0.36516 
0.36481 

0.36447 
0.36412 

0.36377 

9.96294 
9.96  289 
9.96284 
9.96278 
9.96273 

3 

2 

29 

9-59924 
9-59954 
9-59983 
9.60012 
9.60041 

9-63657 
9.63692 
9.63726 
9-63  76i 
9-63796 

0.36343 
0.36308 
0.36274 
0.36239 
0.36204 

9.96267 
9.96262 
9  96256 
9  96251 
9.96245 

35 
34 
33 
32 
_1L 
80 

3 

27 
26 

30 

3i 

32 
33 
34 

9.60070 
9.60099 
9.60  128 
9-60157 
9.60186 

9  •  63  830 
9-63865 
9.63899 

9.63934 
9.63968 

0.36  170 
0.36  135 
0.36  101 
0.36066 
0.36032 

9  .  96  240 
9-96234 
9  96  229 
9-96223 
9.96  218 

9 

12 

i- 

41 
42 
43 
44 

9.60215 
9.60244 
9.60273 
9.60302 
9  60331 

9  64003 
9.64037 
9.64072 
9.64  106 
9.64  140 

0-35  997 
0-35  963 
0.35  928 

0.35894 
0.35  860 

9.96  212 
9.96207 
9.96  201 
9.96  196 
9.96  190 

25 
24 
23 

22 
21 

~w 

19 

i! 

9-60359 
9.60388 
9.60417 
9.60446 
9-60474 

9.64I75 
9.64209 

9-64243 
9.64278 
9.64312 

0-35825 
0-35  79i 
o.35  757 
0.35  722 
0.35688 

9.96  185 

9.96  179 
9.96174 

9.96  168 
9.96  162 

i* 
iti 

49 

9.60503 
9.60532 
9.60  561 
9.60589 
9.60618 

9  64  346 
9-64381 
9.64415 
9.64449 
9.64483 

0-35654 
0.35619 

0-35  585 
o.3555i 
0-35  5i7 

9.96  157 
9.96151 
9.96146 
9.96  140 
9-96I35 

15 
14 
13 

12 
II 

lo" 

I 

60 

5i 

52 
53 

54 

9.60646 
9-60675 
9.60  704 
9.60  733 
9.60  76. 

9-64517 
9  6^552 
9-64586 
9.64620 
9.64654 

0-35  483, 
0.35448 
0.354J4 
0.35380 
0-35  346 

9.96129 
9.96  123 
9.96  1  18 

9.96  112 

9.96  107 

11 

% 

59 

9.60  789 
9.60818 
9.60846 
9.60875 
9.60903 

9.64688 
9.64722 
9.64756 
9.64790 
9.64824 

0.35312 
0.35  278 

0.35  244 
0.35210 
0.35  176 

9.96  ioi 

9.96095 
9.96090 

9  .  96  084 
9.96079 

5 
4 
3 

2 

I 

~TT 

00 

9-60931 

9.64858 

0-35  142 

9.96073 

L.  Cos. 

d. 

L.  Cotg. 

c.d. 

L.  Tang. 

L.  Sin. 

d. 

t 

Prop.  Pts. 

66°     j 

TABLE  IV. 


24° 

t 

L.  Sin. 

d. 

L.  Tang. 

c.d. 

L.  Cots. 

L.  Cos. 

d. 

Proi 

>. 

Pts. 

0 

2 

3 

4 

9.60931 
9.60960 
9.60988 
9.61  016 
9.61045 

29 
28 
28 
29 
28 

9.64858 
9  .  64  892 
9  .  64  926 
9  .  64  960 
9.64994 

34 

34 
34 
34 
34 

0-35  J42 

0.35  108 

0-35  074 
0.35  040 
0.35006 

9.96073 
9.96067 
9  .  96  062 
9  .  96  056 
9  .  96  050 

6 
5 
6 
6 

• 

60 

CQ 

I 

3 

.1     3 

2       6 

4 

•* 

S3 

i:I 

I 
I 

9 

9.61073 
9.61  101 
9.61  129 
9.61  158 
9.61  186 

28 
28 
29 
28 
28 

9  .  65  028 
9  .  65  062 
9  .  65  096 
9.65  130 
9.65  164 

34 
34 
34 
34 
33 

0.34972 
0.34938 
0.34904 
0.34870 
0.34836 

9-96045 
9-96039 
9.96034 
9  .  96  028 
9  .  96  022 

6 
5 
6 
6 

55 
54 
53 
52 
5i 

.3    10 
•4    13 

•5    17 

.6     20 

•  7    23 

2 

.6 

.0 

:i 

99 
13.2 

16  5 
19.8 
23.1 

10 

ii 

12 
«3 

'4 

9.61  214 
9.61  242 
9.61  270 
9.61  298 
9.61  326 

28 
26 
28 
28 
28 

9.65  197 
9.65231 
9.65  265 
9.65299 
9-65  333 

34 
34 
34 

34 

0.34803 
0-34  769 
0-34735 
0.34701 
0.34667 

9.96017 
9.96011 
9.96005 
9.96000 
9-95994 

6 
6 

5 

6 
5 

50 

8 

4J 

.8    27 
•9    30 

.2 

.6 
i 

26.4 
29.7 

>9 

II 

11 

»9 

9-6i  354 
9.61  382 
9.61  411 
9-61  438 
9.61  466 

28 
29 

27 
28 
28 

9.65  400 

9.65434 
9.65  467 
9.65  501 

34 
34 
33 

34 

0.34634 
0.34600 
0.34566 
0-34533 
0-34499 

9.95988 
9.95982 
9-95977 
9-95971 
9-95965 

6 
S 

6 

6 

45 
44 
43 
42 

41 

.1 

.2 

•3 

•4 

t 
I 
ii 
i; 

••2 

•    8 

ll 

.6 
1-5 

20 

21 
22 

23 

24 

9.61  494 
9.61  522 
9-6i  550 
9-6i  578 
9.61  606 

28 
28 
28 
28 
28 

9-65  535 
9.65  568 
9.65  602 
9.65  636 
9.65  669 

33 
34 
34 
33 

0.34465 
0.34432 
0.34398 
0.34364 
0-34331 

9.95960 
9  95954 
9-95948 
9  95942 
9  95937 

6 
6 
6 

5 

6 

40 

39 

38 

8 

j 

•9 

1 

2C 
2; 

7-4 
>-3 

11 

3 

3 

29 

9-61634 
9.61  662 
9.61  689 
9.61  717 
9-6i  74? 

28 
27 
28 
28 
28 

9-65  703 
9-65  736 
9.65  770 
9-65  803 
9-65837 

33 
34 
33 
34 

0.34297 
0.34264 
0.34230 

0-34  197 
0.34  163 

9-95  931 
9.95925 
9.95920 

9.959H 
9.95908 

6 
5 

6 
6 
6 

35 
34 
33 
32 
3i 

.1 

.2 

•  3 

4 

T 

18 
2.8 

ii 

1  .2 

30 
3* 

32 
33 
34 

9.61  773 
9.61  800 
9.61  828 
9.61  856 
9.61  883 

27 
38 
28 
27 
28 

9.65  870 
9.65  904 

9.65937 
9.65971 
9.66004 

34 
33 
34 
33 

0.34  130 
0.34096 
0.34063 
0.34029 
0.33996 

9.95902 

9.95897 
9.95  891 
9.95885 
9.95879 

5 

6 
6 
6 
5 

30 

27 
26 

1 

.9 

I 
I 
I 
2 
2 

*-o 
5.8 
?.6 

2.4 

5-2 

8 
8 

1  39 

9.61  911 
9.61  939 
9.61  966 
9.61  994 

9.62  O2I 

28 
27 
28 
27 
38 

9.66038 
9.66071 
9.66  104 
9.66  138 
9.66  171 

33 
33 
34 
33 

0-33  962 
0.33929 
0.33  896 
0.33862 
0.33829 

9-95873 
9.95868 
9.95  862 
9.95856 
9-95  850 

5 

6 
6 
6 

g 

25 
24 

23 

22 
21 

.1 

.2 

«7 
2.7 
r  4 

,40 

4i 

42 

43 

44 

9.62  049 
9.62076 
9.62  IO4 
9.62  131 
9.62  159 

27 
28 
27 
28 

9.66204 
9.66238 
9.66  271 
9.66304 
9.66337 

34 
33 
33 
33 

0.33  796 
0-33  762 
0.33729 
0.33696 
0.33663 

9-95  844 
9-95  839 
9.95833 
9.95827 
9.95821 

5 

6 
6 
6 
g 

20 

19 
18 

11 

•3 
•4 

'-7 

I 

I 
I 

n 

0.8 

Ii 

8-9 

9 

;i 

49 

9.62  186 
9.62  214 
9.62  241 
9.62268 
9  .  62  296 

28 
27 
27 
28 

9.66371 
9  .  66  404 
9.66437 
9.66470 
9.66503 

33 
33 
33 
33 

0.33  629 
0.33  596 
0-33  563 
0-33  530 

0-33497 

9  958i5 
9-95  810 
9.95  804 
9  95798 
9  95  792 

5 

6 
6 
6 
6 

15 
14 
13 

12 
II 

.8 
•9 

2 
2 

6 

1.6 
4-  3 

5 

150 
5i 
52 
53 
54 

9.62323 
9.62350 
9.62377 
9-62405 
9.62432 

27 
27 
28 

9-66537 
9.66570 
9.66603 
9.66636 
9  .  66  669 

33 
33 
33 
33 

0-33463 
0.33430 
0-33  397 
0.33  364 
0-33331 

9.95  786 
9-95  78o 
9-95  775 
9-95  769 
9-95  763 

10 

I 

.1        C 
.2        1 

•3        1 

•4     J 

•5    : 

>.6 

.2 

.8 
'-4 

;o 

0-5  1 
1.0  ; 

15 
2.0 

25 

P 

11 

59 

9-62459 
9.62486 

9-62513 
9  62  541 
9.62568 

37 

98 

9.66702 

9-66735 
9.66  768 
9.66801 
9-66834 

33 
33 

33 
33 

0.33298 
0.33265 
0-33  232 
0-33  199 
0.33  166 

9-95  757 
9-95  75i 
9  95  745 
9  95  739 
9  95  733 

6 
6 
6 
6 

5 
4 
3 

2 

o    : 

i  ; 

9     I 

;  -t> 

|:S 

1  4 

30 

35 
4.0 

45 

60 

9  62595 

7 

9.66867 

o  33  133 

9  95  728 

0 

L.  Cos. 

d. 

L.  Cotsr. 

c.d. 

L.  Tang. 

L.  Sin. 

d. 

f 

Pro 

P- 

Pts. 

65° 

u 

JUAKl  1  ±1 

MS  L 

*    blJNJL,    I 

.AJMJ 

NJi,    lAJNlj 

r£,JN  1    AJN  1 

)  l^U 

1AJN< 

jUJMl, 

C1 

rc-     5 

25° 

t 

L.Sin. 

d. 

L.  Tang. 

c.  d. 

L.  Cotg. 

L.  Cos. 

d. 

Proj 

>.] 

Pts. 

0 

i 

2 

3 
4 

9-62595 
9.62  622 
9.62  649 
9.62  676 
9.62  703 

27 
27 
27 
27 
27 

9.66867 
9  .  66  900 

9.66933 
9.66966 
9.66999 

33 
33 
33 
33 
33 

0-33  133 
0.33  ioo 
0.33067 

0.33034 
0.33001 

9-95  728 
9-95  722 
9-95  7i6 
9-95  7io 
9-95  704 

6 
6 
6 
6 
6 

60 

59 
58 

H 

3 
•  i      3 

2       6 

3 

33 

3-2 

6.4 

i  *<> 

I 

9 

9.62  730 
9.62  757 
9.62  784 
9.62  81  i 
9.62  838 

27 
27 
27 
27 
27 

9.67032 
9  .  67  065 
9.67098 
9.67131 
9.67  163 

33 
33 
33 
33 
33 

0.32968 

0.32935 
0.32  902 
0.32869 
0.32837 

9.95698 
9.95692 
9-95^36 
9.95  680 
9-95674 

6 
6 
6 
6 
6 

55 
54 
53 
52 
5i 

•3     9 
•4    13 
•5    l6 
.6    19 
•7   23 

•9 

.2 
.  I 

»1 

12.8 

16.0 
19.2 
22.4 

10 

ii 

12 
13 

14 

9.62865 
9.62  892 
9.62  918 

9-62945 
9.62972 

27 
26 
27 
27 
27 

9.67196 
9.67229 
9.67262 
9.67295 
9.67327 

33 
33 
33 
32 

0.32804 
0.32  77i 
0.32  738 
0-32  705 
0.32673 

9.95  668 
9-95  663 
9.95657 
9-9565I 
9  95645 

S 

6 
6 
6 

6 

50 

49 
48 

47 
46 

.8   26 
.9   29 

•4 
•7 

2C.6 
2§.8 

«7 

15 

1  6 

Is7 

19 

9-62999 
9.63026 
9.63052 
9.63079 
9.63  106 

87 

26 
27 
27 

27 

9-67360 

9.67393 
9.67426 
9.67458 
9.67491 

33 

33 
33 
33 

0.32  640 
0.32  607 

0.32574 
0.32  542 
0.32509 

9-95639 
9.95633 
9-95627 
9.95621 
9-956I5 

6 
6 
6 
6 
6 

45 
44 
43 
42 
41 

.1 

.2 

3 
•4 

3 

: 

t 

1C 

i: 

5.7 

11 

>.8 
SI   ' 

20 

21 
22 
23 

24 

9-63  133 
9-63  159 
9.63  186 
9.63213 
9  63239 

26 
27 
27 
26 

27 

9.67524 
9.67556 
9.67589 
9.67  622 
9.67654 

32 
33 
33 
33 

0.32476 
0.32444 

0.324" 

0.32378 
0.32  346 

9.95609 
9-95603 
9-95  597 
9-95591 
9  95585 

6 
6 
6 
6 
6 

40 

i 

:1 

•9 

i* 

2 
2^ 

l'| 

[.6 

L-3 

3 

3 

29 

9.63266 
9.63292 
9.63319 
9.63345 
9.63372 

26 
27 
26 
27 
26 

9.67687 
9.67719 
9.67752 

9-67785 
9.67817 

3« 
33 
33 
32 

0.32313 
0.32  281 
0.32248 
0.32  215 
0.32  183 

9-95579 
9  95  573 
9  95567 
9  95  56i 
9  95  555 

6 
6 
6 
6 
g 

35 
34 
33 
32 
3i 

.1 

.2 

•3 

A 

I 

T( 

* 

2.6 

\l 

5    A 

80 

3i 
32 
33 
34 

9-63398 
9-63425 
9-63451 
9.63478 

9-63504 

27 
26 
27 
26 

l%% 

9.67915 
9.67947 
9.67980 

33 
33 
33 
33 

0.32  150 
0.32  118 
0.32085 
0.32053 
0.32020 

9  95  549 
9-95  543 
9-95  537 
9-9553' 
9-95  525 

6 
6 
6 
6 

6 

30 

11 

2 

i 

.9 

i; 

i 

2( 
2, 

3-°. 

ii 

D.8 

5-4 

P 
9 

39 

9-63  S31 
9.63557 
9-63583 
9.63610 
9-63  636 

26 

26 
27 

2« 

26 

9.68012 
9  .  68  044 
9.68  077 
9  68  109 
9.68  142 

32 
33 
32 
33 

0.31  988 
0.31  956 
0.31  923 
0.31  891 
0.31  858 

9.95519 
9-95  513 
9  95  507 
9-95500 
9-95494 

6 
6 

7 
6 
5 

25 
24 
23 

22 
21 

.1 

.2 

< 

7 

3.7 

1-4 

40 

4i 
42 
43 
44 

9.63  662 
9.63  689 

9-63  715 
9.63  741 
9.63  767 

27 
26 
26 
26 

9.68174 
9.68206 
9  .  68  239 
9.68271 
9.68303 

32 
33 
33 
33 

0.31  826 
0.31  794 
0.31  761 
0.31  729 
0.31  697 

9.95488 
9.95482 
9.95476 
9-95  470 
9.95464 

6 
6 
6 
6 

g 

20 

11 
\l 

•3 

:! 

•7 

t 
t 

Z.I 

2.8 

5-5 

\.2 

t-9 

9 
9 

49 

9-63  794 
9.63820 
9-63  846 
9.63872 
9.63898 

26 
26 
96 
26 
26 

9-68336 
9.68368 
9.68400 
9.68432 
9.68465 

33 
32 

32 
33 

0.31  664 
0.31  632 
0.31  600 
0.31  568 
0.31  535 

9.95458 
9-95452 
9-95  446 
9-95  440 
9-95434 

6 
6 
6 
6 

15 
14 
13 

12 
II 

.8 
•9 

5 

I6 
&3 

5 

50 

5i 

52 
53 
54 

9.63924 
9.63950 
9.63976 
9.64002 
9.64028 

26 
26 
26 
26 
26 

9-68497 
9-68529 
9.68561 
9.68  593 
9.68626 

33 
33 
33 
33 

0.31  503 
0.31471 

0.31  439 
0.31  407 

0.31  374 

9.95427 
9.95421 

9.95415 
9-95  409 
9.95403 

6 
6 

10 

1 
I 

.1       C 
.2       1 

3     i 

•4     a 

i     3 

>.6 
.2 

.8 

•4 
.0 

0-5 
t.o 

i-5 

2.0 
25 

55 
56 

P 

59 

9-64054 
9.64080 
9.64  106 
9  64  132 
9.64  158 

26 
26 
26 
26 
26 

9.68658 
9.68690 
9.68  722 
9.68  754 
9.68  786 

33 

33 
32 

32 

0.31  342 
0.31  310 
0.31  278 
0.31  246 
0.31  214 

9-95  397 
9-95  39i 
9-95384 
9.95378 
9  95372 

5 
4 
3 

2 

6     3 

i  : 

9     5 

6 

.2 

.8 
4 

30 

35 
4.0 

4  5 

60 

9.64  184 

9.68818 

0.31  182 

9-95  366 

0 

L.  Cos. 

d. 

L.  Cotg. 

c.d. 

L.  Tang. 

L.  Sin. 

d. 

9 

Proi 

). 

Pts. 

64° 

6                                                            TABLE  IV. 

26°                                   ll 

t 

L.  Sin. 

d. 

L.  Tang. 

.d. 

L.  Cotg. 

L.  Cos. 

d. 

60- 

9 

JL 

55 
54 
53 
52 
5i 

Prop.  Pts. 

0 

i 

2 

3 
4 

9.64  184 

9.64  2IO 
9.64236 
9.64  262 
9.64288 

26 
26 
26 
26 
25 
26 

86 

26 
26 
25 
26 
36 

25 

26 
26 

25 

26 

25 

26 

25 
26 
25 
26 
25 
26 

«S 
26 
«5 
25 
26 

25 
25 

36 

25 
25 

25 
26 
25 

25 
25 

•3 
25 

26 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
24 
25 
25 
25 
25 
25 

9.68818 
9.68850 
9.68882 
9.68  914 
9.68946 

32 
32 
32 
32 
32 
32 
32 
32 
32 

0.31  182 
0.31  150 
0.31  118 
0.31  086 
0.31  054 

9-95366 
9-9536o 
9-95354 
9-95  348 
9-95341 

6 
6 
6 

6 
6 
6 
6 

7 
6 

6 
6 
6 

7 
6 

6 
6 
7 
6 
6 
6 

7 
6 
6 

6 

7 
6 
6 

7 
6 

6 
6 

7 
6 
6 

7 
6 
6 
7 
6 

6 
7 
6 
7 
6 

6 
7 
6 
6 
7 
6 
7 
6 
6 
7 
6 

7 
6 
6 

7 

& 

:i  I 

•3     9 
.4    12 

•I    l6 
.6    19 

.7     22 

[9  28 
.1 

.2 

•3 
•4 

:i 

i 

•9 
.1 

.2 

•3 
•4 

i 
:I 

•9 
.1 

.2 

•3 
•4 

:i 

:I 

-9 
.1     ( 

.2 

•3     ' 

•4     - 

i  ; 

i  : 

•9     < 

i  j    31 

2     3-1 
4     6.2 

o     9-3 
.8   12.4 

•o  1^.5 
.2   18.6 

.4    21.7 

11$ 

*• 

\i 

S:| 

15-6 
18.2 

20.8 

,3,1! 

as 

2-5 
5-0 
7-5 

10.  0 

12.  5 

15.0 
17.5 

20.  o 

22.5 

N 

:i 

ll 

12.0 

!t.i 

19.2 

21.6 
7          « 

)-7    0.6 
1.4     1.2 
z.i     1.8 

B.8      2.4 

J-S     3-0 

J-2       3-6 

^9     4-2 

j-6     4-8 
>-3     5-4 

1 
I 

9 

IT 

ii 

12 

13 

H 

11 

17 

18 
19 

21 
22 

23 

24 

3 
3 

29 

9-643I3 
9-64339 
9.64365 
9.64391 
9.64417 

9.68978 
9.69010 
9  .  69  042 
9.69074 
9.69  106 

0.31  022 
0.30990 
0.30958 
0.30926 
0.30894 

9-95335 
9-95329 
9-95  323 
9-953I7 
9-95310 

9.64442 
9.64468 
9.64494 
9.64519 
9.64545 

9-69  138 
9.69  170 
9.69202 

9-69234 
9.69  266 

32 
32 
32 
32 

0.30862 
0.30830 
0.30  798 
0.30  766 
0.30734 

9-95304 
9-95  298 
9.95  292 
9  95286 
9  95279 

60 

11 
8 

9.6457I 
9.64596 
9.64622 
9.64647 
9.64673 

9.69298 
9.69329 
9.69361 

9.69393 
9.69425 

3« 

3* 
32 
32 
32 

3i 
32 
32 
32 
3i 
32 
32 
3» 
32 
32 

3* 
32 
3» 
32 
32 

3» 

33 

3« 

32 
3« 
32 
3» 
32 
3* 
32 

3» 
3i 
32 
3» 
32 

3i 
3* 
32 
3» 
3» 
32 
3» 
3* 
3» 
32 

O.30  702 
0.30671 
0.30639 
0.30607 
0.30575 

9-95273 
9-95  267 
9.95261 

9-95254 
9.95248 

45 
44 
43 
42 

41 

9.64698 
9.64  724 
9.64749 

9.64775 
9.64  800 

9-69457 
9.69488 
9.69520 
9.69552 
9.69584 

0.30543 
0.30512 
0.30480 

o  .  30  448 

0.30416 

9.95242 
9  95236 
9.95229 
9  95223 
9.95217 

40 

P 
ll 

9.64826* 
9.64851 
9.64877. 
9.64902 
9.64927 

9.69615 
9.69647 

9-69679 
0.69  710 
9.69742 

0.30385 
0.30353 
0.30321 
0.30290 
0.30258 

9-952II 
9.95204 
9.95  198 
9-95  192 
9  95  185 

35 
34 
33 
32 

i 

1 

25 
24 
23 

22 
21 

w 

10 

ii 

\l 

30 

3i 

32 
33 
34 

9.64953 
9.64978 
9.65003 
9.65  O29 
9-65054 

9.69774 

9-69805 
9.69837 
9.69868 
9.69900 

0.30  226 

0.30  195 
0.30  163 
0.30  132 
0.30  100 

9  95  179 
9-95  173 
9  95  167 
9-95  160 
9-95  '54 

5 

9 

39 

9.65079 
9.65  104 
9.65  130 
9.65  155 

9.65  180 

9.69932 
9.69963 

9.69995 
9.70026 
9.70058 

0.30068 

0.30037 
0.30005 
0.29974 
0.29942 

9.95  148 
9-95  HI 
9-95  135 
9  95  129 
9  95  122 

40 

4» 
42 

43 
44 

9-65205 
9-65230 

9.65  255 
9.65  281 
9.65306 

9.70089 

9.70  121 
9.70152 
9.70184 
9.70215 

0.29  911 
0.29879 

0.29848 

0.29  816 
0.29  785 

9-95  "6 
9  95  "0 
9  95  103 
9.95097 
9.95090 

9 

.a 

49 

9-6533I 
9-65356 

9-  653*i 
9.65406 

9.65431 

9.70247 
9.70278 
9.70309 
9.70341 
9.70372 

0.29  753 
0.29  722 
0.29  691 
0.29  659 
o  .  29  628 

9.95084 
9.95078 
9.95071 
9.95065 
9-95059 

15 

14 
13 

12 
II 

60 

51 
52 
53 
54 

9.65456 
9.65  481 
9.65506 
9  65531 
9  65  556 

9.70404 
9-70435 

9  .  70  466 
9.70498 
9-70529 

0.29596 
0.29565 
0.29  534 
0.29502 
0.29471 

9-95052 
9.95046 
9-95039 
9-95033 
9.95027 

10 

I 

i? 

12 

59 

E 

9.65580 
9  65605 
9  65  630 

9-65655 
9.65  680 

9.70560 
9.70592 
9.70623 
9.70654 
9  •  70  685 

0.29440 
o  .  29  408 
0.29377 
0.29  346 
0.29315 

9  95  020 
9  95014 
9  95007 
9.95001 
9  94  995 

s 

4 
3 

2 
I 

~cr 

9.65  705 

9.70717 

o  .  29  283 

9.94988 

L.  Cos. 

d. 

L.  Cotg. 

c.d 

L.  Tang. 

L.  Sin. 

d. 

t 

Prop.  Pts. 

63°                                        | 

LOGARITHMS  OF  SINE,  COSINE,  TANGENT  AND  COTANGENT,  ETC.     57 


27° 

9 

L.  Sin. 

d. 

L.  Tang. 

.d. 

L.  Cotgr. 

L.  Cos. 

d. 

p 

rop 

,1 

>ts. 

0 

I 

2 

3 
4 

9.65  705 
9.65  729 
9-65754 
9-65  779 
9.65804 

24 

25 
25 
25 

24 

9.70717 
9.70748 
9.70779 
9.70  810 
9.70841 

3» 
3» 
3» 
3« 

o  .  29  283 
0.29252 

0.29  221 
O.29  190 
0.29  159 

9.94988 
9.94982 

9-94975 
9.94969 
9.94962 

6 
7 
6 

7 
6 

60 

5? 
58 

Ii 

.1 

.2 

32 

I 

2 

/) 

31 

ii 

1 
I 

9 

9.65828 
9-65853 
9-65878 
9.65902 
9.65927 

25 
25 
24 
25 
25 

9.70873 
9  .  70  904 

9.70935 
9.70966 
9.70997 

3i 
3« 
31 
3« 
31 

0.29  127 
0.29096 
0.29065 
0.29034 
0.29003 

9.94956 
9.94949 
9-94943 
9.94936 
9-94930 

7 
6 

7 
6 

55 
54 
53 
52 
5i 

•3 
•4 

:! 

•  7 

9 

12 

16 
19 

22 

6 
8 

0 
2 

4 

9-3 
12.4 

21.7 

10 

ii 

12 
13 
14 

9-65952 
9.65976 
9  .  66  ooi 
9.66025 
9.66050 

24 

23 
24 
25 
25 

9.71  028 
9.71059 
9.71090 

9.71   121 
9-71   153 

3« 
31 
31 
32 
31 

0.28972 
0.28941 
0.28910 
0.28879 
0.28  847 

9.94923 
9.94917 
9.94911 
9.94904 
9.94898 

6 
6 

7 
6 

50 

3 
t? 

.8 
•9 

3 

6 
8 

3 

24.8 
27.9 

0 

15 
10 

\l 

19 

9-66075 
9.66099 
9.66  124 
9.66  148 
9-66173 

24 
25 
24 
«5 

24 

9.71   184 
9.71  2l| 
9.71  246 
9.71277 
9.71308 

3* 
3i 
3* 
3* 

0.288l6 
0.28  785 

0.28  754 
0.28  723 
0.28  692 

9.94891 
9.9488; 
9.94878 
9.94871 
9.94865 

6 
7 
7 
6 

45 
44 
43 
42 

41 

i 

.2 

•3 
•4 

i 

I 

$ 

12 

;s 

.0 
.O 
.0 
.0 
.0 
!  o 

20 

21 
22 
23 

24 

9.66  197 

9.66  221 
9.66246 
9.66270 
9.66293 

24 
25 
24 
25 

9.71339 
9.71370 
9.7I40I 

9.7I43I 
9.71  462 

3« 
3» 
30 
3» 

0.28  661 
o  .  28  630 
0.28599 
o  28  569 
0.28  538 

9.94858 
9.94852 
9.94845 
9-94839 
9.94832 

6 
7 
6 

7 

6 

40 

fs 

11 

i 

•9 

21 

24 
2; 

.O    ' 
[0 

r.o 

25 
20 

3 

29 

9-66319 

r«is 
IM 

24 
25 
24 
«4 

9$ 

9.7I493 
9-71  524 
9.7I555 
9.71586 
9.7I6I7 

3i 
3« 
3» 
3* 

0.28  507 
0.28476 
0.28445 
0.28414 
0.28383 

9  .  94  826 
9.94819 
9.94813 
9.94806 
9-94799 

7 
6 

7 

7 
6 

35 
34 
33 
32 
3i 

.1 

.2 

•3 

A 

a 
2 

5 
7 
10 

5 

•5 

.0 

•5 

o 

«4 

2  4 
4.8 

& 

30 

3i 

32 
33 
34 

9.66441 
9.6646? 
9.66489 
9-66513 
9-66537 

24 
24 
24 
24 
25 

9  71  648 
9.71679 
9.71  709 
9.71  740 
9.71  771 

3* 
30 

3i 
3i 

0.28352 
0.28  321 
0.28  291 
0.28  260 
0.28  229 

9-94793 
9.94786 
9.94780 

9-94773 
9.94767 

7 
6 

7 
6 

so 

29 
28 
27 
26 

''I 
•9 

12 
15 
17 
20 
22 

•5 

.0 

-5 
.0 

-S 

12.  0 

14  4 
16.8 
192 

21.6 

II 

9 

39 

9.66  562 
9.66586 
9.66610 
9.66634 
9.66658 

24 
24 
24 
24 

9.71  802 
9-71833 
9.71863 
9.71894 
9.71  925 

3i 
30 
3* 
3» 

0.28  198 
0.28  167 
0.28  137 
0.28  1  06 
0.28075 

9.94760 
9  94753 
9-94  747 
9.94740 

9  94  734 

7 
6 
7 
6 

25 
24 
23 

22 
21 

.1 
? 

23 

a 

40 

41 

42 
43 
44 

9.66  682 

9  .  66  706 
9.66731 

9-66755 
9.66779 

24 
25 
24 
24 

9.7I955 
9.71  986 
9.72017 
9  .  72  048 
9.72078 

3i 
3» 
3i 
30 

o  .  28  045 
0.28  014 
0.27983 
0.27952 
0.27922 

9.94727 
9.94720 
9.94714 
9-94  707 
9.94700 

7 
6 

7 

7 

| 

20 

19 

il 

•3 
•4 

:i 

•  7 

, 

i 

5,9 
9.2 

il 

3 

4J 

49 

9.66803 
9.66827 
9.66851 
9  66875 
9.66  899 

24 
24 
•  24 

24 

9.72  109 
9.72  140 
9.72170 
9  .  72  201 

9.72231 

3i 
30 
3i 
30 

o  27891 
0.27  860 
0.27  830 
0.27  799 
0.27  769 

9.94694 
9.94687 
9  .  94  680 
9.94674 
9.94667 

7 
7 
6 
7 

15 
14 
13 

12 
II 

.8 
•  9 

i 

2 
7 

8.4 

0.7 

6 

50 

5i 
52 
53 

54 

9.66  922 
9  .  66  946 
9.66970 
9.66994 
9.67018 

24 
24 
24 
24 

9  .  72  262 
9.72293 
9-72323 
9.72354 
9.72384 

3i 
30 

3« 
30 

0.27738 
0.27  707 
0.27677 
0.27646 
0.27616 

9  .  94  660 

9-94654 
9.94647 
9.94640 
9-94634 

7 
6 
7 
7 
6 

10 

I 

.1 

.2 

•3 
•4 

c 

i 

2 
2 

3 

-7 
•4 
.  i 
.S 
•5 

0.6 

1.2 

1.8 
2.4 
3-0 

I 

s 

59 

9.67042 
9.67066 
9.67090 
9.67113 
,9.67137 

24 
24 
23 
24 

9-72415 
9.72445 
9.72476 
9.72506 
9.72537 

30 
3i 
30 
3i 

0-27585 

0-27555 
0.27524 
.0.27494 
0.27463 

9.94627 
9  .  94  620 
9.94614 
9.94607 
9.94600 

7 
6 
7 
7 

5 
4 
3 

2 

I 

:1 

•9 

4 
4 

1 

.2 

1 

•3 

3-6 
4.2 
4-8 
5-4 

60 

9.67  161 

9.72  567 

0.27433 

9-94593 

0 

L.  Cos. 

d. 

L.  Cotg. 

c.d 

L.  Tang 

L.  Sin. 

d. 

t 

] 

VO] 

>• 

Pts. 

62° 

g                                                            TABLE  IV. 

28° 

1 

I.  Sin. 

d. 

L.  Tang. 

c.d. 

L.  Cotg. 

L.  Cos. 

d. 

Prop.  Pts. 

0 

9.67  161 

9.72567 

0.27433 

9-94593 

00 

I 

2 

3 
4 

9.67  185 
9.67208 
9.67232 
9.67256 

23 
24 
24 
24 

9.72598 
9.72  628 
9.72659 
9.72  689 

31 

3° 

31 

0.27402 
0.27372 
0.27341 
0.27311 

9-94587 
9-9458o 
9-94573 
9-94567 

7 
7 
6 

.1 

2 

31 
3-1 

6   2 

30 

I 

9.67280 
9.67303 

23 
24 

9.72  720 
9.72750 

30 

0.27  280 
0.27  250 

9-9456o 
9-94553 

7 

55 
54 

•  3 
-4 

9-3 
12.4 

9.0 
12.0 

I 

9.67327 
9.67350 

23 

9.72  780 
9.72811 

3i 

0.27  220 
0.27  189 

9.94546 
9.94540 

6 

53 

S2 

•i 

•5 

15.0 

18.0 

9 

9-67374 

24 

9.72841 

31 

0.27159 

9-94533 

7 

•  7 

21 

.7 

21  .0 

10 

9.67398 

9.72872 

0.27  128 

9.94  526 

50 

.8 

24 

.8 

24.0 

ii 

12 

9.67421 

9-67445 
9.67468 

24 
23 

9.72902 
9-72932 
9-72963 

3° 
30 
3* 

0.27098 
0.27068 
0.27037 

9-94  5J9 
9-94513 
9.94506 

7 
6 

7 

49 
48 

•9 

27 

9 

27.0 

*4 

9.67492 

23 

9-72993 

30 

0.27007 

9-94499 

7 

46 

39 

!<> 

9.675I5 

9.73023 

0.26977 

9.94492 

45 

.1 

2.9 

16 

9-67539 

9-73054 

31 

0.26  946 

9-94485 

7 

44 

.2 

5-8 

17 
18 

9.67562 
9.67  586 

24 

9.73084 
9-73ii4 

30 

0.26916 
0.26886 

9-94479 
9.  94  472  , 

7 

43 
42 

•3 
•4 

& 

19 

9.67609 

23 
24 

9-73  144 

3° 

0.26856 

9-94465 

7 

4i 

•5 

14-5 

20 

9.67633 

9  73  175 

0.26  825 

9-94458 

40 

7 

17-4 
20  7 

21 

9.67656 

9-73205 

0.26  795 

9-94451 

39 

'I 

22 

9.67680 

9.73235 

30 

0.26  765 

9-94445 

38 

Q 

23.2 
26  i 

23 

24 

9.67703 
9.67726 

23 
24 

9-73265 
9.73295 

30 
3 

0-26735 
0.26  705 

9-94438 
9-94431 

7 
7 

•y 

25 

9.67750 

9.73326 

0.26  674 

9-94424 

35 

34             »» 

26 
29 

9.67773 
9-67796 
9.67820 
9.67843 

23 

24 
23 
23 

9.73356 
9.73386 
9.73416 
9-73446 

3° 
30 
30 
30 

0.26  644 
0.26  614 
0.26  584 
0.26554 

9.94417 
9.94410 
9.94404 
9-94397 

7 
7 
6 

7 

34 
33 
32 

.1 
.2 

•  3 

A 

7-2 
o  6 

V6 

6.9 

92 

30 

9.67866 

9-73476 

0.26  524 

9-94390 

30 

12.0 

II.  j 

32 
33 
34 

9.67890 
9.67913 
9.67936 
9  67959 

23 
23 
23 
23 

9-73507 
9-73537 
9-73567 
9-73597 

31 

30 
30 
30 

0.26  493 
o  .  26  463 
0.26433 
0.26403 

9-94383 
9-94376 
9-94369 
9-94362 

7 
7 
7 
7 

1 

i 

.9 

14.4 

16.8 
19.2 

21.6 

20.7 

% 

9.67982 
9.68006 

24 

9.73627 

30 

0.26373 
0.26343 

9-94355 
9  94  349 

6 

25 
24 

B 

9.68029 
9.68052 

23 
23 

9-73687 
9-73  717 

30 

30 

0.26313 
0.26  283 

9-94342 
9-94335 

7 
7 

23 

22 

33 
2  2 

39 

9.68075 

23 

9-73  747 

30 

0.26253 

9-94328 

7 

21 

.2 

4-4 

40 

9.68098 

9-73777 

o  .  26  223 

9-94321 

20 

-3 

6.6 

41 

9.68  121 

9.73807 

3° 

0.26  193 

9-943H 

7 

19 

.4 

8.8 

42 
43 

9.68  144 
9.68  167 

23 
23 

9.73867 

3° 

30 

0.26  163 
0.26  133 

9-94307 
9.94300 

7 
7 

18 
17 

i 

II.  0 

13.2 

44 

9.68  190 

23 

9.73897 

3° 

0.26  103 

9-94293 

7 

16 

.7 

9 

9.68  213 

24 

9-73927 
9-73957 

30 

0.26073 
0.26043 

9.94286 
9.94279 

7 

15 

•9 

17.6 
19.8 

47 

9.68260 

9-  73  987 

3° 

0.26013 

9-94273 

13 

48 

9.68283 

9.74017 

30 

0.25983 

9  .  94  266 

7 

12 

49 

9.68305 

9.74047 

30 

0.25  953 

9-94259 

7 

II 

7 

6 

50 

9.68328 

9.74077 

0.25923 

9-94252 

10 

.1 

0.7 

0.6 

5' 

9-6835I 

23 

9-74  107 

3° 

0.25893 

9-94245 

7 

9 

.2 

] 

[-4 

1.2 

52 
53 

9-68374 
9.68397 

23 
23 

9.74137 
9.74166 

30 
29 

0.25863 
0.25  834 

9.94238 
9.94231 

7 
7 

7 

-3 
•4 

2.1 
2.8 

1.8 
2.4 

54 

9.68420 

23 
23 

9.74196 

30 

0.25  804 

9.94224 

7 

6 

•5 

3-5 

8-8 

B 

ii 

9.68443 
9.68466 
9.68489 
9-68  512 

23 
23 
23 

9.74226 
9.74256 
9  .  74  286 
9.74316 

30 
30 
30 

0.25  774 
0.25  744 
0.25  714 

0.-25  684 

9.94217 

9.94  2IO 
9.94203 
9.94196 

7 
7 
7 

5 
4 
3 

2 

.b 

i 

•9 

4.2 
4-9 

I'6 
6-3 

3-6 
5-4 

59 

9-68534 

23 

9-74345 

29 

0-25655 

9-94  189 

7 

I 

•» 

00 

9.68557 

9-74375 

0.25  625 

9.94  182 

0 

L.  Cos. 

d. 

L.  Cots. 

c.d. 

L.  Tancr. 

L.  Sin. 

d. 

t 

Prop.  Pts. 

I                                         61° 

LOGARITHMS  OF  SINE,  COSINE,  TANGENT  AND  COTANGENT,  ETC.      r 


29°                                       i 

9 

— 

I 
2 

3 
4 

L.  Sin. 

d. 

L.  Tang. 

c.d. 

L.  Cotg. 

L.  Cos. 

d. 

Prop.  Pts. 

9-68557 
9.68580 
9  68603 
9.68625 
9.68648 

23 
23 

22 
23 
«3 

23 
22 
23 
23 
22 

23 
22 

23 
23 
23 

23 
29 

23 
22 

23 
32 

23 
22 
23 
22 
22 

23 
22 

23 
22 

22 
23 
22 
22 
22 

23 
33 
93 
33 

22 

«3 
32 
33 
33 
33 
33 
32 
22 
22 
23 
32 
22 
22 
22 
22 
22 
23 
22 
22 
23 

9-74375 
9-74405 
9-74435 
9.74465 
9-74494 

3° 
3° 
3° 
29 
30 
30 
29 
30 
30 
30 

29 

30 
30 

29 
30 

30 
29 
3<> 
29 
30 

29 
30 
30 
29 
30 

29 
30 
29 
30 
29 
30 
29 
30 
29 
29 
30 
29 
30 
29 
29 
30 
29 
30 
29 
29 
30 
29 
29 
29 
30 
29 
29 
29 
30 
29 
29 

29 
30 
29 
29 

o  .  25  625 
0  25  595 
0-25565 
0-25  535 
0.25  506 

9.94  182 

9.94175 
9.94  168 
9.94161 
9  94154 

7 
7 
7 
7 
7 
7 
7 
7 
7 
7 
7 
7 
8 
7 
7 
7 
7 
7 
7 
7 
7 
7 
7 
8 

7 
7 
7 
7 
7 
7 

7 

8 

7 
7 
7 
7 
7 
8 
7 
7 
7 
7 
8 

7 

7 

7 
8 

7 
7 
7 

7 
8 

7 
7 
8 

7 

7 
7 

8 

7 

00 

3 

11 

.1 

.2 
•3 

•4 

i 

•9 
.1 

.2 

•3 

:I 
:| 

•9 
.1 

.2 

•3 

•4 

:! 

i 

•9 
.1 

.2 

•3 
•4 

:i 

:l 

•  9 

.1       C 

.2       1 
.3       ' 

.4    : 

•i  : 

:l  \ 

.9    ; 

30 

3° 
6.0 
9.0 

12.0 

I5-° 
18.0 

21.0 
24.0 
27.0 

«9 
2.9 

H 

n.  6 

14.5 
17-4 
20.3 

2J.2 
20.1 

93 

i1 

6.9 

9-2 

"•5 

'3-8 
IO.I 

18.4 
20.7 

99 

2.2 

to 

8.8 

II.  O 

13-2 
15-4 
17.6 
|I9.8 

8         7 

).8     0.7 
.6     1.4 
(.4     2.1 
5.2     2.8 
[.o     3.5 
t.8     4.2 
;.6     4-9 
»-4     5-6 
r.2     6.3 

1 

9 

9.68  671 
9  .  68  694 
9.68  716 
9.68739 
9.68  762 

9.74524 
9^74554 
9-74583 
9.74613 

9.74643 

0.25  476 
0.25  446 
0.25417 
0.25387 
0.25357 

9.94147 
9.94  140 

9-94I33 
9.94  126 
9.94  119 

55 
54 
53 
52 
Si 

10 

ii 

12 
»3 
14 

9.68  784 
9.68807 
9.68829 
9.68852 
9.68875 

9.74673 
9.74702 

9-74732 
9.74762 

9-74791 

0.25327 
0.25  298 
0.25  268 
0.25  238 
0.25  209 

9.94  112 
9.94105 
9.94098 
9.94090 
9.94083 

50 

42 
48 

47 
46 

45 
44 
43 
42 

41 

!2 
!2 

19 

9.68897 
9.68920 
9.68942 
9.68965 
9.68987 

9.74821 
9-74851 
9.74880 
9.74910 
9-74939 

0.25  179 
0.25  149 

O.25  120 
O.25  09O 

0.25  06  1 

9.94076 
9.94069 
9.94062 
9-94055 
9.94048 

20 

21 
22 
23 

24 

9.69010 
9.69032 
9  69055 
9.69077 
9.69  100 

9.74969 
9.74998 
9.75028 
9.75058 
9.75087 

0.25  031 

O.25  002 
0.24972 
0.24942 
0.24913 

9.94041 

9-94034 
9.94027 
9.94020 
9.94012 

40 

39 
38 

1 

35 
34 
33 
32 
3i 

% 

11 

29 

9.69  122 

9.69144 
9.69167 
9.69  189 
9.69212 

9-75  "7 
9-75  H6 
9-75  176 
9-75205 

9-75  235 

0.24883 
0.24854 
0.24824 
0.24795 
0.24765 

9.94005 
9-93998 
9-93991 
9-93984 
9-93977 

n 

31 
32 

33 

34 

9.69234 
9.69256 
9.69279 
9.69301 
9.69323 

9-75  264 
9-75294 
9.75323 
9-75353 
9.75382 

0-24736 

o  .  24  706 

0.24677 
0.24647 

0.24618 

9.93970 
9.93963 
9-93955 
9-93948 
9-93941 

30 

27 
26 

i 

39 

IT 

41 
42 

43 

44 

9-69345 
9.69368 
9.69390 
9.69412 
9-69434 

9-754" 
9-75441 
9-75470 
9-75500 
9.75529 

0.24589 

0.24559 
0.24530 
0.24500 
0.24471 

9-93934 
9.93927 
9.93920 
9.93912 
9-93905 

25 
24 
23 

22 
21 

"20" 

!98 
\l 

9-69456 
9-69479 
9.69501 

9-69523 
9.69545 

9-75558 
9-75588 
9-756I7 
9-75647 
9.75676 

0.24442 

0.24412 

0.24383 
0.24353 
0.24324 

9-93898 
9.93891 
9.93  884 
9.93  876 
9.93869 

9 

8 

49 

9.69567 
9.69589 
9.69  611 
9-69633 
9-69655 

9-75705 
9-75735 
9-75  764 
9-75793 
9-75822 

0.24295 

0.24  265 
0.24236 
0.24207 
0.24  178 

9.93862 
9.93855 
9.93847 
9.93840 

9-93833 

15 
H 
13 

12 
II 

50 

5i 

52 
53 
54 

9.69677 
9.69699 
9.69721 
9  69743 
9  69765 

9-75852 
9.75881 
9.75910 
9-75939 
9.75969 

0.24  148 
0.24  119 
0.24090 
0.24061 
0.24031 

9.93826 
9.93819 
9.93811 
9.93804 
9-93  797 

10 

1 
I 

55 
56 

9 

59 

1ST 

9  69787 
9.69809 
9.69831 
9  69853 
9  69875 

9-75998 
9.76027 
9.76056 
9.76086 
9.76  115 

0.24002 
c  23973 

0.23944 

0.23  914 

0.23885 

9-93  789 
9-93  782 
9-93  775 
9.93  768 
9  93  76o 

5 
4 
3 

2 
I 

"0" 

9  69  897 

9.76  144 

o  23  856 

9-93753 

L.  Cos. 

d. 

L.  Cotg. 

c.d. 

L.  Tang. 

L.  Sin. 

d. 

t 

Prop.  Pts. 

60° 

TABLE  IV* 


i                                         80°                                        1 

t 

L.  Sin. 

d. 

L.  Tang. 

.d. 

L.  Cotg. 

L.  Cos. 

d. 

Prop.  Pis. 

i  ' 
\ 

9.69897 
9.69919 
9.69941 

9  69963 
9.69984 

23 

33 
33 
21 
23 

9.76  144 
9.76  173 

9  .  76  202 
9.76231 
9.76  26l 

39 
29 
29 
30 
29 

39 
29 
29 

«9 
29 

39 

29 
29 
39 
29 
29 
30 
29 
39 
28 
39 
39 
29 
29 
39 
29 
29 
29 
29 
39 
29 
29 
28 
29 
29 
29 
29 
29 
28 
29 
29 
29 
29 
28 
29 
29 
29 
28 
29 
29 
28 
29 
29 
29 
28 
29 
28 
29 
29 
28 

0.23  856 
0.23  827 
o  23  798 
0.23  769 
o  23  739 

9  93  753 
9  93  746 
9  93  738 
9-9373" 
9  93  724 

7 
8 
7 
7 
7 
8 
7 
7 
8 
7 

7 
8 

7 
8 

7 

7 
8 
7 
7 
8 

7 
8 

7 
7 
8 

7 
8 

7 
8 
7 

7 
8 

7 
8 
7 
8 

7 

8 

7 
8 

7 
8 

7 

8 
7 
8 

7 
8 
7 
8 

7 
8 

7 
8 
8 

7 
8 

7 

8 

7 

00 

3 

11 

3< 

:i  I 

•3     9 
.4    12 

:i  !I 

its 

.9  27 
.1 

.2 

•3 

:I 
•i 

•9 
.1 

.2 

•3 
.4 

i 

:J 

•9 
.1 

.2 

•3 
•4 

:i 

i 

•9 
.1     ( 

.2 

•3     '• 
•4      , 
•  5     * 

.6      - 

:2  . 

•9 

>      «« 

o    2.9 
o     j.8 
.0     8.7  1 
.0   ii.  6 
.0  14.5 
.0  17.4 
.0  20.3 
.0  23.2 

.0    20.1 

38 
2.8 

s-6 

1-4 

II.  2 

14  o 
16.8 
19.6 
22.4 
25.2 

sa 

2.2 

ft 

8.8 

II.  0 

13  2 
"5-4 
17.6 
I9.8 

at 
2.1 

4  2 
63 
8.4 
10  5 

12.6 

£2 

18.9 

8          7 

5.8     0.7 
[.6     1.4 

2.4      2.1 
J.2       2.8 

*o    3.5 
^.8     4-2 
C.6     49 
3-4     5-6 
7.2     6.3 

i 
2 

9 
10 
ii 

12 
13 
H 

\l 

11 

19 

21 

22 
23 

24 

9  .  70  006 
9  .  70  028 
9  .  70  050 
9.70072 
9.70093 

33 
32    , 
22 
21 
23 
33 
33 
91 
32 
33 
31 
22 
81 
23 
32 
21 
23 
21 
22 
21 
22 
31 
33 
21 
33 
21 
33 
81 
33 
31 
31 
23 
21 
31 
33 
31 
31 
21 
22 
21 
21 
21 
22 
21 
21 
21 
21 
21 
23 
21 
XX 
21 
21 
21 
21 

9.76290 
9.76319 
9.76348 

9.76377 
9.76406 

0.23  710 
0.23  68  1 
0.23652 
0.23623 
o  23  594 

9  93717 
9  93  709 
9.93702 

9-93695 
9-93687 

9.93680 

9  93673 
9  93  66  J 
9-93658 
9  93  650 

55 
54 
53 
52 
5" 
60" 

3 
2 

9.70115 
9.70137 
9.70159 
9.70  180 
9  .  70  202 

9.76435 
9.76464 
9.76493 
9.76522 
9-7655I 

0.23565 
0.23536 
0.23  507 
0.23478 
0.23449 

9.70224 
9.70245 
9.70  267 
9.70288 
9.70310 

9.76580 
9.76609 
9.76639 
9.76668 
9.76697 

o  .  23  420 
0.23391 
0.23361 
0.23332 
0.23303 

9-93643 
9  93636 
9  93  628 
9-9362I 
9.93614 

45 
44 
43 
42 

4" 

9-70332 
9.70352 
9.70375 
9.70396 
9.70418 

9.76725 

9-76  754 
9-76783 
9.76  812 
9.76841 

0.23275 
0.23  246 
0.23217 
0.23  188 
0.23  159 

9.93606 
9-93599 
9-9359" 
9  93  584 
9  93577 

40 

3 

1 

35 
34 
33 
32 

i 

I 

s 
2 

29 

31 
32 

33 

Jl_ 

$ 

12 

39 

4i 
42 
43 

1  44 

9.70439 
9.70461 
9.70482 
9.70504 
9-70525 

9.76870 
9.76899 
9.76928 

9.76957 
9.76986 

0.23  130 
0.23  101 
0.23072 
0.23043 
0.23  014 

9-93569 
9  93562 
9-93554 
9-93547 
9  93539 

9-70547 
9-70568 
9.70590 
9.70611 
9-70633 

9.77015 

9.77044 
9.77073 
9.77  101 
9.77130 

0.22985 

0.22  956 
0.22  927 
0  .  22  899 
0.22  870 

9-93532 
9-93525 
9-93  5"7 
9-93510 
9-93  502 

9.70654 
9.70675 
9.70697 
9.70718 
9-70739 

9.77159 
9.77188 
9.77217 
9.77246 
9.77274 

0.22  841 
0.22  8l2 
0.22  783 

0.22  754 

0.22  726 

9-93495 
9  93487 
9-9348o 
9-93472 
9  93465 

25 
24 

23 

22 
21 

w 

19 

ii 

9.70761 
9.70782 
9.70803 
9.70824 
9  .  70  846 

9.77303 
9-77332 
9.77361 

9-77390 
9.77418 

0.22  697 
0  .  22  668 
O.22  639 
O.22  6lO 
0.22  582 

9  93457 
9  93450 
9  93442 
9  93435 
9  93427 

4£ 
46 

% 

49 
10" 

5i 

52 
53 
J£ 

9 

55I 

I 

9.70  867 
9.70888 
9.70909 
9.70931 
9.70952 

9-77447 
9.77476 
9-  775°5 
9-77533 
9.77562 

0.22  553 

0.22  524 
0.22495 
0.22  467 
0.22438 

9.93420 
9  93412 
9  93405 
9  93397 
9  93390 

15 
14 

"3 

12 
II 

lo~ 

1 

5 
4 
3 

2 
I 

~0" 

9-70973 
9.70994 
9.71  015 
9.71036 
9.71058 

9-77591 
9.77619 
9-77648 
9.77677 
9.77706 

0  .  22  409 
0  22  381 
0.22352 
0.22  323 
0  22  294 

9  93382 
9  93  375 
9  93  367 
9-93360 
9  93  352 

9.71  079 
9.71  loo 

9.71   121 
9.71   142 

9  7i  I63 

9-77734 
9-77  763 
9.77791 
9.77820 
9.77849 

0.22  266 
0.22237 
O.22  2O9 
0.22  l80 
0.22  151 

9-93344 
9  93337 
9  93329 
9  93322 

9  93314 

9  71  184 

9.77877 

0.22  123 

9  93  3°7 

L.  Cos. 

d. 

L.  Cots 

c.d, 

L.  Tang 

L.  Sin. 

d. 

/ 

Prop.  Pts. 

59°                                        1 

LOGARITHMS  OF  SINE,  COSINE,  TANGENT  AND  COTANGENT,  ETC.     6Y 


31° 

/ 

L.  Sin. 

d. 

L.  Tang. 

c.d. 

L.  Cotg. 

L.  Cos. 

d. 

W 

CQ 

57 
56 

Prop.  Pts. 

0 
I 

2 

3 
4 

9.71  184 
9.71  205 
9.71  226 
9.71247 
9.71  268 

21 
21 
21 
21 
21 
21 
21 
21 
21 
20 
21 
21 
21 
21 
21 
21 
20 
21 
21 
21 
2O 
21 

si 

91 
2O 
81 

21 
20 
21 
21 
20 
21 
20 
21 
20 
21 
20 
21 
21 
20 
20 
21 
20 
21 
20 
21 
2O 
2O 
21 
20 
20 
21 
20 
20 
21 
30 
2O 
M 
90 
30 

9.77877 
9.77906 

9-7793? 
9.77963 

9-77992 

29 
29 
28 
29 
28 

0.22  123 
0.22094 
0.22065 
0.22037 
0.22008 

9-93307 
9.93299 

9-93  291 
9.93284 
9.93276 

8 

8 

7 
8 

7 

8 
8 

7 
8 
8 

7 
8 
8 
7 
8 

8 

7 
8 
8 
7 
8 
8 
7 
8 
8 

7 
8 
8 
8 
7 
8 
8 
8 

7 
8 
g 

8  ' 

.1 

.2 

•3 
•4 

ii 

•9 
.1 

.2 
•3 

:! 
:J 

•9 
.1 

.2 

•  3 
•4 

ii 

•9 
.1 

.2 

.3 

•4 

ii 
ii 

•  9 
.1     < 

.2        ] 

•3     - 

.4    : 

:i  ; 

:i  i 

9    ; 

39 

S* 

,!:i 

H-5 
17  4 
20.3 

li.l 

38 

2.8 

5.6 
8.4 

II.  2 

14-0 

16  8 
19.6 
22.4 
25.2 

ax 
2.1 

X 

8-4 

S:i 
£Z 

18.9 
20 

2.O 

4-0 
6.0 

8.0 

IO.O 
12.  0 
14-0 

16.0 
180 

8          7 

).8    0.7 
1.6     1.4 

J.4      2.1 
J.2      2.8 

^o    3-5 
k8    4-2 
;.6    4.0 

)-4    5-6 

r  2    63 

1 

I 

9 

9.71289 
9.71310 
9-7i33i 
9-71352 
9  7i  373 

9  .  78  020 
9.78049 
9.78077 
9.78  106 
9.78135 

29 
28 
29 

29 
28 

0.21  980 
0.21  951 
0.21  923 
0.21  894 
0.21  865 

9.93269 
9.93261 

9-93253 
9-93246 
9-93238 

55 
54 
53 
52 
51 

"50" 

49 
48 

8 

10 

ii 

12 
'3 

14 

9.71393 
9.71414 

9-7i  435 
9-71  456 
9.71  477 

9-78163 
9.78192 
9  .  78  220 
9.78249 
9.78277 

29 
28 
29 
28 
29 
28 
29 
28 
28 
29 

38 

29 

98 

29 

38 

28 
29 
28 
29 
28 
28 
29 
28 
28 
29 
28 
28 
29 
28 
28 
28 
29 
28 
28 
28 
29 
28 
28 
28 
28 
29 
28 
28 
28 
28 
28 
39 
28 
28 
28 

0.21  837 
0.21  808 
0.21  780 
0.21  751 
0.21  723 

9.93230 
9.93223 
9-932I5 
9-93207 
9-93200 

11 

II 

19 

9.71  498 
9.7i  5*9 
9-71  539 
9.71  560 
9.71  581 

9.78306 
9-78334 
9-78363 
9.78391 
9.78419 

0.21  694 

0.21  666 

0.21  637 
O.2I  609 

0.21  581 

9-93  192 
9-93  l84 
9-93  177 
9-93  169 
9.93  161 

45 

44 
43 
42 

* 

9 

11 

20 

21 
22 
23 

24 

9.71  602 
9.71  622 
9-71  643 
9.71664 
9.71  685 

9.78448 
9-78476 
9-78505 
9.78533 
9-78562 

0.21  552 
O.2I  524 

0.21  495 

0.21  467 
0.21  438 

9  93  154 
9-93  H6 
9-93  138 
9  93  131 
9  93  123 

3 

27 
28 

3 

3i 
32 
33 

34 

9.71  705 
9.71  726 
9.71  747 
9.71  767 
9.71  788 

9.78590 
9.78618 
9.78647 
9.78675 
9.78  704 

0.21  410 
O.2I  382 
0-21353 
0.21325 
0.21  296 

9-93  "5 
9.93  1  08 
9.93  loo 
9.93092 
9.93084 

35 
34 
33 
32 

* 

% 
2 

25 
24 

23 

22 
21 

~w 

19 
18 

\l 

9.71  809 
9.71  829 
9-71850 
9.71  870 
9.71  891 

9-78732 
9.78760 

9-78789 
9-73817 
9-78845 

0.21  268 
0.21  240 
O.2I  211 
O.2I  183 
0.21  155 

9.93077 
9.93069 
9.93061 

9.93053 
9.93046 

9 

II 

39 

9.71  911 
9.71932 
9.71952 
9-7i  973 
9.71994 

9-78874 
9.78902 
9.78930 
9-78959 
9-78987 

0.21  126 
0.21  098 
0.21  07O 
O.2I  041 
0.21  013 

9-93038 
9.93030 
9.93022 
9.93014 
9.93007 

40 

4i 
42 
43 
44 

9.72014 
9.72034 
9.72055 
9.72075 
9.72096 

9.79015 

9-79043 
9.79072 
9.79  loo 
9.79  128 

0.20985 
0.20957 
0.20928 
O.2O  9OO 
0.20§72 

9.92999 
9.92991 
9.92983 
9-92976 
9  .  92  968 

S 
S 

49 

9.72  116 
9.72  137 
9.72  157 

9.72177 
9.72  198 

9-79I56 
9-79  185 
9-792I3 
9.79241 
9.79269 

o  .  20  844 

0.20815 
0.20  787 
0.20759 

o  20  731 

9  .  92  960 
9.92952 
9.92944 
9.92936 
9.92929 

15 
14 
J3 

12 
II 

ICT 

z 

50 

5i 
52 
53 

54 

9.72  218 
9.72238 
9.72259 
9.72279 
9-72299 

9-79297 
9.79326 

9-79354 
9.79382 
9.79410 

0.20  703 
O.2O  674 
0  .  2O  646 
O.206l8 
0.20  590 

9.92921 
9.92913 
9.92905 
9.92  897 
9.92  889 

55 
56 

9 

59 

9.72320 
9.72340 
9.72360 
9.72381 
9.72401 

9-79438 
9-79466 
9-79495 
9.79523 
9-79551 

0.20  562 
0.20534 
0.20  505 
0.20477 
0.20449 

9.92881 
9.92874 
9.92866 
9-92858 
9.92  850 

5 
4 
3 

2 
I 

~0" 

60 

9.72421 

9-79579 

O.2O  421 

9.92842 

L.  Cos. 

d. 

L.  Cotg. 

c.d. 

L.  Tang. 

L.  Sin. 

d. 

/ 

Prop.  Pts. 

58° 

TABLE  IV. 


32° 

9 

L.  Sin. 

d. 

L.  Tang. 

c.d. 

L.  Cotg. 

L.  Cos. 

d. 

p 

roi 

).  ] 

Pts. 

0 

I 

2 

3 
4 

9.72421 
9.72441 
9.72461 
9.72482 
9.72502 

20 

20 

21 
2O 

20 

9-79579 
9.79607 

9.79635 
9.79663 
9.79691 

28 
28 
28 
28 
98 

0.20421 
0.20393 

0.20  365 
0.20337 
0.20309 

9.92842 
9.92834 
9.92  826 
9.92818 
9.92  810 

8 
8 
8 
8 

GO 

59 
58 

% 

.1 

2 

z 
2 

9 

38 
2.8 

<:  6 

i 

I 

9 

9.72522 
9-72542 
9-72562 
9.72582 
9  .  72  602 

20 
20 
2O 
2O 
2O 

9-79  719 
9-79  747 
9-79  776 
9.79804 
9-79832 

28 
29 
28 
28 
28 

O.2O  28l 
0.20253 
0.20224 
0.20  196 

0.20  168 

9.92  803 

9-92  795 
9.92  787 

9-92  779 
9-92  77i 

8 
8 
8 
8 
g 

55 
54 
53 
52 

•3 
•4 

•  7 

14 
17 

2C 

•5 
-4 

.T, 

:>  -u 
8.4 

II.  2 
14-0 

16.8 
19.6 

12 
13 

H 

9.72622 

9.72643 
9-72663 
9.72683 
9-72703 

21 
2O 
20 
20 
2O 

9  .  79  860 
9.79888 
9.79916 

9-79944 
9.79972 

28 
28 
28 
28 
28 

0.20  140 
0.20  112 
0.20084 
0.20056 
0.20028 

9-92  763 
9.92755 
9.92747 

9-92  739 
9.92  731 

8 
8 
8 
8 
3 

1 

% 

.8 
•9 

% 

.2 

a 

22.4 
25.2 

7 

19 

9.72723 

9-7?  743 
9.72763 
9.72783 
9.72803 

20 
20 
20 
20 
2O 

9.80000 
9.80028 
9  .  80  056 
9.80084 

9.80  112 

28 
28 
28 
28 
28 

0.20000 
0.19972 
0.19944 
O.I9  916 
0.19888 

9.92  723 
9.92  715 

9  92  707 
9.92699 
9.92691 

8 
8 
8 
8 
3 

45 
44 
43 
42 

.1 
.2 

•3 
•4 

1 

2 

i 

1C 

•7 

.4 

iis 

20 

21 
22 
23 

24 

9-72823 

9-72843 
9.72863 
9-72883 
9.72902 

2O 
20 
20 
19 

9.80  140 
9.80168 
9-80  195 
9.80223 
9.80251 

28 
27 
28 
28 
28 

0.19  860 
0.19832 
0.19805 
0.19777 

o.  19  749 

9.92  683 
9.92675 
9.92667 
9.92659 
9.92651 

8 
8 
8 
8 
3 

40 

1 

1 

•9 

I* 

2] 
2; 

*-9 
1.6 

t-3 

27 
28 

29 

9.72922 
9-72942 
9.72962 
o  .  72  982 
9.73002 

20 
20 
2O 
2O 

9.80279 
9.80307 
9.80335 
9.80363 
9.80391 

28 
28 
28 
28 
28 

0.19  721 
0.19  693 
0.19  665 
0.19637 
0.19  609 

9-92643 
9-92635 
9.92627 
9.92  619 
9.92  611 

8 
8 
8 
8 
3 

35 
34 
33 
32 

.1 

.2 

•  3 

4 

-. 

2 
4 

( 

J 

II 
!.I 

I   2 

H 

30 
2.0 
4.0 

6.0 
8.0 

30 

32 
33 
34 

9.73022 
9.73041 
9.73061 
9.73081 
9.73101 

20 
20 
20 

9.80419 
9.80447 
9.80474 

9  .  80  502 
9.80530 

28 
27 
28 
28 
28 

0.19581 

0.19553 
0.19  526 
0.19498 
0.19470 

9.92603 

9-92  595 
9.92587 

9-92579 
9.92571 

8 
8 

8 
8 
3 

30 

29 
28 

% 

•9 

1C 

i: 
i, 
K 
I. 

11 

1-7 
3.8 

5-9 

IO.O 
12.  0 
14.0 

16.0 
18.0 

9 

9 

39 

9.73121 
9.73140 
9-73  160 
9.73180 
9.73200 

J9 

20 
20 
20 
10 

9.80558 
9.80586 
9.80614 
9  .  80  642 
9.80669 

28 
28 
28 
27 
28 

0.19  442 
0.19414 
0.19  386 
0.19358 
0.19331 

9-92563 
9-92555 
9.92  546 

9-92  538 
9-92530 

8 

9 
8 
8 
3 

25 
24 
23 

22 
21 

.2 

[  .  C 

9 

40 

41 
42 

43 
44 

9.73219 
9.73239 
9-73259 
9.73278 
9.73298 

20 
20 

20 

9.80697 
9-80725 
,9.80753 
9.80  781 
9.80808 

28 
28 
28 
27 
28 

0.19303 
0.19275 
0.19247 
0.19  219 
0.19  192 

9.92522 
9.92514 
9-92506 
9.92498 
9-92490 

8 
8 
8 
8 
3 

20 

19 
18 

17 
16 

•  3 

•  7 

i 

i 

\l 

?-5 
i.4 

3-3 

1:1 

4-5 
54 
6-3 

49 

9-733I8 
9-73337 
9-73357 
9-73377 
9  73396 

19 
20 
20 

'9 

9.80836 
9.80864 
9.80892 
9.80919 
9.80947 

28 
28 
27 
28 
28 

o.  19  164 
0.19  136 
0.19  108 
0.19081 
0.19053 

9.92482 

9-92473 
9.92465 

9-92457 
9.92449 

9 

8 
8 
8 
3 

15 
14 
13 

12 
II 

.8 
•9 

i 

i 

5-2 
7-1 

8 

21 

7 

W 

53 
54 

9-734i6 
9-73435 
9-73455 
9-73474 
9-73494 

20 
20 

9.80975 
9.81  003 
9.81  030 
9.81  058 
9.81  086 

28 
27 
28 
28 

0.19025 
0.18997 
o.  18  970 
0.18  942 
0.18  914 

9.92441 

9-92433 
9.92425 
9.92416 
9  .  92  408 

8 
8 

9 
8 
3 

10 

I 

.1 

.2 

•3 
•  4 

< 

t 

D.8 

1.6 
2.4 
3-2 

*-s 

o-7 

2.1 

2.8 

3-5 

59 

9  73513 
9-73533 
9  73552 
9  73572 
9  73591 

2O 

20 
19 

9.81  113 
9.81  141 
9.81  169 
9.81  196 
9!8i  224 

28 
28 
27 
28 
28 

0.18887 
0.18859 
0.18831 
0.18804 
o.  18  776 

9.92  400 
9.92  392 
9.92384 

9-92367 

8 
8 
8 

9 
3 

s 

4 
3 

2 
I 

9 

t 
( 

\.t> 

\\ 

7-2 

4.2 
4-9 

1.1 

!()() 

9  73  6n 

9.81  252 

0.18  748 

9-92359 

0 

L.  Cos. 

d. 

L.  Cotg. 

c.d. 

L.  Tang. 

L.  Sin. 

d. 

f 

1 

To 

1>. 

Pte. 

57° 

LOGARITHMS  OF  SINE,  COSINE,  TANGENT  AND  COTANGENT,  ETC. 


33° 

/ 

L.  Sin. 

d. 

L.  Tang. 

c.d. 

L.  Cotg. 

L.  Cos. 

d. 

Prop.  Pte. 

0 

I 

2 

3 
4 

9.73611 
9.73630 

9-73650 
9.73  669 
9.73689 

19 

20 

*9 

30 
19 

'9 
20 

19 

J9 

20 

19 
19 
2O 

19 

*9 

20 
»9 

J9 
19 
19 
20 
19 
*9 
J9 
J9 
20 
»9 
»9 
>9 
«9 
19 
19 
J9 

*9 

X9 
19 
19 
»9 
i9 
19 
*9 
*9 
19 
i9 
*9 
19 
»9 
19 
18 

19 
i> 
19 
19 
19 
18 

19 
19 
19 
18 

*9 

9.81  252 
9.81  279 
9.81  307 
9-8i  335 
9.81  362 

«7 
28 
28 
27 
28 
28 
27 
28 
27 
28 

0.18  748 
0.18  721 
0.18  693 
0.18665 
0.18638 

9  92359 
9-92351 
9-92343 
9-92335 
9.92326 

8 

8 
8 

9 
8 

8 
8 

9 

8 
8 
8 

9 

8 
8 
9 
.  8 
8 
8 
9 
8 

& 

9 
8 
8 
9 
8 
8 

9 
8 
8 

9 

8 
8 

9 
8 

9 
8 
8 

9 
8 

9 

8 

8 

9 
8 

9 

8 

9 

8 

9 
8 

9 
8 

9 
8 

9 
8 

9 
8 

9 

60 

3 

11 

a 

.1        2 

I    I 

.4     II 

•5    H 
.6    16 

•7    19 

.8     22 

•9    25 
.1 

.2 

•3 

:! 

i 

•9 

.1 

.i 

.3 
.4 

;i 
i 

•9 
.1 

2 

•  3 

:1 

i 

•9 

.1        C 
.2        1 

•3     2 

•4     2 

:l  1 

•  7     * 

:5  ^ 

8        37 

.8    2.7 
.6     c.4 
.4     8.1 

.2     10.8 

o  J3.5 
.8   16.2 
.6   18.9 
.4  21.6 

.2    24.3 

90 
2.O 

1° 

6.0 
8.0 

IO.O 
12.  0 
14-0 
IO.O 

18.0 
«9 

5:1 
M 

95 
H.4 

J3.3 
15.2 
17.1 

it 

1.8 
36 

5-4 
7.2 

9.o 
10.8 

12   6 

14.4 

16.2 

9           • 

>-9     0.8 
.8     1.6 
.7      2.4 
.6     3.2 
•5     4.o 
•4     4.8 
•3     5-6 
.2     6.4 
.1     72 

I 

I 

9 

9.73708 
9.73727 

9-73747 
9.73766 

9-73785 

9-8i  390 
9.81418 

9.81445 
9  8i473 
9.81  500 

0.18  610 
0.18  582 

0.18555 
0.18  527 
0.18  500 

9.92318 
9.92310 
9.92302 
9.92293 
9-92285 

55 
54 
53 
52 
5i 

10 

ii 

12 
13 

14 

9.73805 
9.73824 
9.73843 
9.73863 
9.73882 

9.81  528 
9  81  556 
9.81  583 
9.81  611 
9.81  638 

28 
27 
28 
27 
28 

0.18472 
0.18444 
0.18417 
0.18389 
0.18  362 

9.92277 
9.92269 
9.92  260 
9  92252 
9.92244 

50 

3 
3 

15 
10 

\l 

19 

,20 

21 
22 

23 

24 

9.73901 
9.73921 
9-73940 
9  73959 
9.73978 

9.81  666 
9.81  693 
9.81  721 
9.81  748 
9.81  776 

27 
28 
27 
28 
27 
28 
27 
28 
27 
28 
27 
28 
27 
28 
27 
28 
27 
28 
27 
27 
28 
27 
28 
27 
27 
28 
27 
28 
27 
27 
28 
27 
27 
28 
27 
27 
28 
27 
27 
27 
28 
27 
27 
27 
28 

0.18334 
0.18307 
0.18  279 
0.18252 
0.18  224 

9-92235 
9.92227 
9.92  219 

9.92  211 
9  .  92  202 

45 
44 
43 
42 
41 
40" 

i 

9-73997 
9.74017 
9.74036 

9-74055 
9.74074 

9.81  803 
9.81  831 
9.81858 
9.81  886 
9.81913 

0.18  197 
0.18  169 
0.18  142 
0.18  114 

0.18087 

9.92  I94 

9.92  186 

9-92  177 
9.92  169 
9.92  161 

3 

i  27 
28 
29 

9-74093 
9.74"3 
9-74132 

9-74  IS1 
9.74170 

9.81941 
9.81  968 
9.81  996 
9.82023 
9.82051 

0.18  059 
0.18032 
0.18004 

0.17977 
0.17949 

9-92  152 
9.92  144 
9.92  136 
9.92  127 
9.92119 

35 
34 
33 
32 
31 
30 

29 
28 
27 
26 

30 
31 

32 
33 
34 

9.74189 
9.74208 
9.74227 
9.74246 
9.74265 

9.82078 
9.82  106 
9.82  133 
9.82  161 
9.82  1  88 

0.17922 

0.17894 
0.17867 
0.17839 
0.17  812 

9.92  in 

9.92  102 

9  .  92  094 
9  .  92  086 
9.92077 

9 

i? 

39 

9.74284 

9  -743°3 
9.74322 

9-74341 
9.74360 

9.82  215 
9.82243 
9.82  270 
9.82  298 
9.82325 

0.17785 
0.17757 
0.17730 
0.17  702 
0.17675 

9.92069 
9.92060 
9-92052 
9.92044 
9-92035 

25 
24 
23 

22 
21 

W 

il 

40 

4i 
42 

43 
44 

9-74379 
9-74398 
9.744I7 
9.74436 
9-74455 

9.82352 
9.82380 
9.82407 

9-82435 
9  .  82  462 

0.17  648 
0.17  620 
0.17593 
0.17565 
0.17538 

9.92027 
9.92  018 
9.92010 
9  .  92  002 
9-91  993 

3 

s 

49 

9-74474 
9-74493 
9-74512 

9-74531 
9-74549 

9.82489 
9.82517 
9.82544 
9.82571 
9.82599 

0.17511 
0.17483 
0.17456 
0.17429 
0.17401 

9.91  985 
9.91  976 
9.91  968 
9-9i  959 
9-9i  95i 

15 
H 
13 

12 
II 

50 

5i 
52 
53 
54 

9.74568 

9-74587 
9.74606 
9.74625 
9.74644 

9.82  626 
9.82653 
9.82681 
9.82  708 
9-82  735 

0.17374 
0.17347 
0.17319 
0.17  292 
0.17265 

9.91  942 

9-9i  934 
9.91  925 
9.91  917 
9.91  908 

10 

I 
I 

5 
4 
3 

i 

55 
56 

H 

59 

9.74662 
9.74681 
9.74700 
9.74719 
9-74737 

9.82  762 
9.82  790 
9.82  817 
9.82  844 
9.82871 

0.17238 

0.17  210 
0.17  183 
O.I7I56 
O.I7  129 

9.91  900 
9.91  891 

9-91  f3 
9.91  874 
9.91  866 

00 

9-74756 

9.82899 

0.17  ioi 

9-91  857 

0 

L.  Cos. 

d. 

L.  Cotg. 

c.  d. 

L.  Tang. 

L.  Sin. 

d. 

/ 

Prop.  Pts. 

56° 

TABLE  IV. 


34° 

/ 

L.  Sin. 

d. 

L.  Tang. 

c.d. 

L.  Cotg. 

L.  Cos. 

d. 

Pro] 

0. 

Pte. 

0 
I 

2 

3 

4 

9-74756 
9-74775 
9-74794 
9.74812 
9-74831 

X9 
X9 
18 

X9 

9.82899 
9.82  926 

9-82953 
9.82980 
9.83008 

27 
27 
27 
28 
27 

0.17  101 
0.17074 
0.17047 
0.17  020 
0.16992 

9.91  857 
9.91  849 
9.91  840 
9.91  832 
9.91  823 

8 

9 
8 

9 

8 

58 

11 

g 

.1       2 
2       e 

8 

.8 
6 

27 
2.7 

r    A 

i 
I 

9 

9.74850  J 
9.74868 
9-74887 
9.74906 
9.74924 

18 
X9 
X9 
18 
to 

9-83035 
9.83062 
9-83089 
9-83117 
9-83  144 

27 
27 
28 
27 
27 

0.16  965 
0.16938 
0.16911 
0.16883 
0.16856 

9.91815 
9.91  806 
9.91  798 
9.91  789 
9.91  781 

9 

8 

9 
8 

55 
54 
53 

5i 

•3     8 
.4    ii 

•  5    14 
.6    16 
.7    19 

•4 

.2 
.0 

.8 
.6 

11 

10.8 
13-5 

16.2 

18.9 

10 
ii 

12 
13 

9-74943 
9.74961 
9.74980 

9-74999 
9.75017 

18 

X9 
x8 

X9 

9-83  171 
9.83  198 
9-83225 
9-83252 
9.83280 

27 
27 
27 
28 
27 

0.16  829 
0.16  802 
0.16775 
0.16  748 
0.16  720 

9.91  772 
9-91  763 
9-9i  755 
9.91  746 
9.91  738 

9 

9 
8 

9 
8 

50 

8 
3 

.8     22 

•9    25 

•4 

.2 
i 

21.6 

24.3 

(6 

19 

9-75036 
9-75054 
9.75073 
9.75091 

9  75  "° 

18 

X9 
18 

X9 

18 

9-83307 
9-83334 
9-83361 
9.83388 

9-83415 

27 
27 

27 

0.16693 
0.16666 
0.16639 
0.16  612 
0.16  585 

9.91  729 
9.91  720 
9.91  712 
9.91  703 
9.91  695 

9 
8 

9 
8 

45 
44 
43 
42 

41 

.1 

.2 

•3 
•4 

•1 

2 
c 
i 
i 
1C 

i; 

>..6 

>-4 
•'6 

20 

21 
22 
23 

24 

9-75  128 
9-75  147 
9-75  165 
9-75  184 
9.75202 

X9 

18 

X9 
18 

9.83442 
9.83470 
9-83497 
9-83524 
9-83551 

28 

2? 
27 
27 

0.16  558 
o.  16  530 
0.16503 
0.16476 
o  16  449 

9.91  686 
9.91  677 
9.91  669 
9.91  660 
9.91  651 

9 

9 

8 

9 
9 

40" 

P 

H 

.0 
•9 

ij 

2< 

2; 

>-6 

$.2 

>.8 
M 

29 

9-75221 
9.75239 
9-75258 
9-75276 
9.75294 

18 

X9 
18 
18 

9.83578 
9.83605 

9-83659 
9.83686 

27 
27 
27 
27 
27 

0.16422 
0.16395 
0.16368 
0.16341 
0.16  314 

9.91  643 
9.91  634 
9.91  625 
9.91617 
9.91  608 

9 

9 
8 

9 

35 
34 
33 
32 

.1 

.2 

-3 

A 

'    1 

[9 

7  6 

80 

32 
33 
34 

9-753I3 
9-75.331 
9-75350 
9-75368 
9  75386 

18 

X9 
18 
18 

ig 

9-83  7i3 
9.83  740 
9-83  768 
9-83  795 
9.83822 

27 
28 
27 
27 
27 

0.16287 
0.16260 
0.16  232 
0.16  205 
0.16  178 

9-9i  599 
9-91  59i 
9.91582 

9.91573 
9  -9i  S6? 

9 
8 
9 
9 
8 

30 

27 
26 

•9 

( 
I 

i; 

i 

i 

>-5 
1-4 
5-3 

5-2 

7-1 

P 

3^ 
39 

9  75405 
9-75423 
9  75441 
9-75459 
9  75478 

18 
18 
18 

X9 
18 

9-83849 
9-83876 
9-83903 
9-83930 
9.83957 

27 
27 

27 

27 

0.16  151 
0.16  124 
0.16097 
0.16  070 
o.  16  043 

9-91  556 
0  91  547 

9.91  530 
9.91  521 

9 
9 
8 

9 

25 
24 

23 

22 
21 

.1 

.2 

IS 

[.8 
z.6 

40 

42 
43 
44 

9-75496 
9-755I4 
9-75533 
9-75551 
9-75  569 

18 

18 
18 

18 

9-83984 
9.84  on 
9.84038 
9.84065 
9  .  84  092 

27 
27 
27 
27 

o.  16  016 
0.15989 
0.15  962 

0.15935 
0.15  908 

9.91512 
9.91  504 

9-91  495 
9.91  486 
9.91  477 

9 
8 
9 
9 

9 
3 

20 

19 
18 

17 
16 

-3 

•  4 

i 

K 

I 

5-4 
7-2 
?-o 

D.8 
2.6 

47 
48 
49 

9-75587 
9-75605 
9  75  624 
9.75642 
9.75660 

It 

18 
18 
18 

9.84  119 
9  .  84  146 
9.84173 
9  .  84  200 

9.84227 

«7 
27 
27 
27 

0.15881 
0.15854 
0.15  827 
0.15  800 
0.15  773 

9.91  469 
9.91  460 

9-91  451 
9.91  442 

9-91433 

9 
9 

9 

9 
8 

15 
14 
13 

12 
II 

•9 

9 

8 

50 

5* 
52 
53 
54 

9.75696 
9-757H 
9-75733 
9-75751 

18 

18 

X9 
18 

18 

9.84254 
9.84280 
9.84307 

9.84334 
9.84361 

27 
27 
27 

0.15  746 
o.  15  720 
0.15  693 
o.'is  666 
0.15639 

9.91  425 
9.91  416 
9.91407 
9.91  398 
9.91  389 

9 
9 
9 

9 
8 

10 

I 

.1        C 

.2        1 

•3     '< 

.4    : 

•5       4 

>  9 

.8 

5-7 
\-6 

[•5 

0.8  i 
i  6 
2  4 
3-3 

11 

56 
59 

9.75769 
9.75787 
9.75805 
9-75823 
9-75841 

z8 
18 
x8 
18 
18 

9.84388 
9.84415 
9.84442 
9.84469 
9.84496 

27 
27 
27 

0.15  612 
0-15585 
0.15558 
O.I553I 
0.15  504 

9.91  381 
9.91  372 
9-91  363 
9-91  354 
9-91  345 

9 
9 
9 
9 

5 

4 
3 

2 

'.7     * 

:5  1 

-4 
>-3 

r.2 

;.i 

4.S 

I6 
6.4 

7.2 

60 

9-75  859 

9.84523 

0.15477 

9-9i  S36 

9 

0 

L.  Cos. 

d. 

L.  Cotg. 

c.d. 

L.  Tancr. 

L.  Sin. 

d. 

t 

Pro 

>• 

Pis. 

55° 

LOGARITHMS  OF  SINE,  COSINE,  TANGENT  AND  COTANGENT,  ETC. 


85° 

t 

L.  Sin. 

<1. 

L.  Tang. 

c.d. 

L.  Cotg. 

L.  Cos. 

d. 

Proj 

.1 

?ts. 

0 

I 

2 

3 
4 

9-75859 
9.75877 
9.75895 
9.75913 
9-75931 

18 
It 

18 

18 
18 

9-84523 
9-8455o 
9.84576 
9  .  84  603 
9.84630 

27 
26 
27 
27 
27 

o.i5477 
0.15450 
0.15424 

0.15397 
0.15370 

9-91  S36 
9.91  328 
9.91319 
9.91  310 
9.91  301 

8 
9 
9 
9 
9 

60 

52 
58 

H 

2 
.1        2 
.2        5 

7 

-7 
4 

26 
2.6 

C.2 

I 

I 

9 

9-75949 
9.75967 
9.75985 
9/76003 
9.76021 

18 
18 
18 
18 

18 

9-84657 
9.84684 
9.84711 
9.84738 
9.84764 

27 
27 
27 
26 
27 

0-15343 
0.15  316 
0.15  289 
0.15  262 
0.15  236 

9.91  292 
9.91  283 
9.91  274 
9.91  266 
9.91  257 

9 
9 
8 

9 
9 

55 
54 
53 
52 
5i 

•3     8 
.4    10 

:7  Is 

.1 

.8 

•5 

.2 

•9 

7.8 

10.4 
I3.0 

i£.6 
18.2 

10 
ii 

12 
13 

14 

9.76039 
9.76057 
9.76075 
9.76093 
9  .  76  1  1  1 

18 
18 
18 
18 
18 

9.84791 
9.84818 
9.84845 
9  .  84  872 
9.84899 

27 
27 
27 
27 
26 

0.15  209 
0.15  182 

0-15  155 
0.15  128 
0.15  101 

9.91  248 
9.91  239 
9.91  230 

9.91  221 
9.91  212 

9 
9 
9 
9 

50 

49 
48 
47 
46 

.8     21 

•9    24 

.6 
•3 

i 

20.8 

23  4 
8 

11 

\l 

19 

9.76129 
9.76146 
9.76164 
9.76  182 
9  .  76  200 

»7 
18 
18 

18 
18 

9.84925 
9.84952 
9.84979 
9.85  006 
9-85033 

27 
27 
27 
27 
26 

0-15075 
0.15  048 

0.15  021 
0.14994 
0.14967 

9.91  203 
9.91  194 
9.91  185 
9.91  176 
9.91  167 

9 
9 
9 
9 

45 
44 
43 
42 

4i 

.1 

.2 

•3 
•4 

\ 

1 

t 
< 

.8 
1-6 

5-4 

r-2 
)  o 

)O 

20 

21 
22 

23 

24 

9.76  218 

9.76236 
9-76253 
9.76271 
9.76289 

18 

»7 

18 
18 

18 

9-85059 
9.85086 

9-85  "3 
9.85  140 
9.85  166 

27 
27 
27 
26 

0.14941 
O.I49I4 
0.14887 
0.14860 
0.14834 

9.91  158 
9.91  149 
9.91  141 
9.91  I32 
9.91  123 

9 
8 

9 
9 

40 

li 
II 

i 

•9 

i: 

i. 
K 

.5 

1.6 

11 

11 

27 
28 
29 

9.76307 
9.76324 
9.76342 
9.76360 
9.76378 

«7 

18 

18 

18 

9-85  193 

9.85  220 

9-85  247 

9.85273 
9.85300 

27 
27 
26 

27 

0.14807 
O.I4  780 

0-14753 
0.14727 
0.14  700 

9.91  114 
9.91  IO5 

9  91  096 
9.91087 
9.91  078 

9 
9 
9 
9 

35 
34 
33 
32 
3i 

.1 

.2 

•3 

A 

t7 
t-7 
J-4 

Ij 

30 

3i 

32 
33 
34 

9-76395 
9.76413 
9.76431 
9.76448 

9  .  76  466 

18 
18 
17 
18 
18 

9-85327 
9-85354 
9-85380 
9.85407 
9-85434 

27 
26 
27 
27 

26 

O.I4  673 
0.14  646 
0.14620 

0.14593 
0.14566 

9.91069 
9.91  060 
9.91051 
9.91042 
9  91  033 

9 

9 
9 
9 

30 

22| 
11 

1 

i 

•9 

i 
i 
i 
i 

*-5 

D.2 

1  1 

36 

5-3 

9 
9 

39 

9.76484 
9.76501 
9  76519 
9.76537 
9.76554 

17 

18 
18 

17 
18 

9.85460 
9-85487 
9-855I4 
9.85540 

9-85  567 

27 
27 
26 
27 

0.14540 

O.I45I3 
0.14486 
0.14460 
0-14433 

9.91023 
9.91014 
9.91  005 
9.90996 
9.90987 

9 

9 
9 
9 

25 

24 
23 

22 
21 

.1 

.2 

zo 

I.O 

2.O 

40 

41 
42 
43 
44 

9.76572 
9.76590 
9.76607 
9-76625 
9  .  76  642 

18 

»7 

18 

*7 
x8 

9  85594 
9.85  620 
9.85647 
9.85674 
9.85  700 

26 
27 
27 
26 

0.14406 
0.14380 
0-14353 

o.  14  326 

0.14300 

9.90978 
9.90969 
9.90960 
9.90951 
9.90942 

9 

9 
9 
9 

20 

19 

ii 

•3 
-4 

:i 

.7 

30 
4.0 

|-o 
5.0 

7.0 

9 
9 

49 

9.76  660 
9.76677 
9-76695 
9.76712 
9.76730 

17 
18 
»7 
18 

9.85  727 

9-85  754 
9.85  780 
9-85807 
9-85834 

27 

27 
26 
27 
27 

ofi 

0.14273 
0.14246 

0.14220 

0.14193 

0.14  166 

9  90933 
9.90924 
9.90915 
9.90906 
9  .  90  896 

9 
9 
9 
9 
xo 

15 
14 
13 

12 
II 

.8 
•9 

9 

8.0 

30 

8 

50 
5i 

52 
53 

_5!_ 

9.76747 
9.76765 
9.76782 
9.76800 
9.76817 

18 

17 

18 

»7 
18 

9.85860 
9.85887 

9-859I3 
9.85940 
9.85967 

27 
26 

27 
27 

0.14  140 
0.14113 
0.14  087 
0.14060 
0.14033 

9.90887 
9.90878 
9  .  90  869 
9  .  90  860 
9  .  90  85  1 

9 
9 
9 
9 
9 

10 

I 

.1       C 
.2        1 

•3     2 
•4      2 

'}  A 

>-9 

•'•I 

r-5 

0.8 
1.6 
2.4 
32 

4'S 

55 
56 

12 

59 

9-76835 
9-76852 
9.76870 
9.76887 
9.76904 

17 

18 

»7 
17 
18 

9  85993 
9  .  86  020 
9.86046 
9.86073 
9.86  loo 

27 
26 

27 

27 

0.14007 
0.13980 

0-13954 
0.13927 
o  13  900 

9.90842 
9.90832 
9.90823 
9.90814 
9  90805 

9 
xo 
9 
9 
9 

5 
4 
3 

2 

I 

:?  i 

.8     I 

.9    i 

•  4 
•3 

.2 

.1 

4-8 
56 
6.4 

7  2 

GO 

9.76922 

9.86  126 

o  13874 

9  90796 

9 

0 

L.  Cos. 

d. 

L.  Cotgr. 

c.d. 

L.  Tang. 

L.  Sin. 

d. 

t 

Pro] 

>• 

Pfs. 

54° 

66 


TABLE  IV. 


36° 

1 

L.  Sin. 

(1. 

L.  Tang. 

C.<1. 

L.  Cot!?. 

L.  Cos. 

<1. 

Pro] 

>. 

Pts. 

0 

I 

2 

3 
4 

9.76922 
9  76939 
9-76957 
9.76974 
9.76991 

17 

18 
J7 
»7 

18 

9.86  126 
9-86  153 
9.86179 
9.86206 
9.86232 

27 
26 
27 
26 

27 

0.13874 
0.13847 

0.13  821 

0.13794 

0.13  768 

9.90  796 
9.90787 
9.90777 
9.90768 
9.90  759 

9 

10 

9 
9 

GO 

3 

11 

3 
I       2 
2        C 

7 

•7 

| 

26 

2.6 

IT    2 

I 
I 

9 

9.77009 
9.77026 

9-77043 
9.77061 
9.77078 

17 
17 

18 

»7 
17 

9.86  259 
9.86285 
9.86312 
9.86338 
9.86365 

26 
27 
96 
27 
27 

o.  13  741 
0.13  7i5 
0.13688 
0.13  662 
0.13635 

9.90750 
9.90  741 
9.90731 
9.90  722 
9.90  713 

9 

9 

x> 

9 
9 

55 
54 
53 
52 
5i 

•3     * 
.4    ic 

*  \l 

•7    iS 

.1 
.8 
•5 

.2 

9 

fl 

10.4 

ill 

10 
ii 

12 
13 

'4 

9-77095 
9.77»2 
9.77130 

9-77  H7 
9.77164 

»7 
18 
»7 
*7 
*7 

9  86392 
9.86418 

9.86445 
9.86471 
9.86498 

26 
27 
26 
27 
26 

0.13608 
0.13582 

0.13555 
0.13529 
0.13  502 

9.90704 
9-90694 
9.90685 
9.90  676 
9.90667 

9 

10 

9 
9 
9 

50 

3 
% 

.8     21 

.9  24 

.6 
•3 

20.8 

23.4 

18 

III 

17 
18 

19 

9.77181 
9.77199 
9.77216 

9.77233 
9-77250 

18 
17 
«7 
17 
18 

9-86524 
9.86551 
9.86577 
9.86603 
9.86630 

27 
26 
26 
27 
2<> 

0.13476 
0.13449 
0.13423 
0.13397 
0.13370 

9.90657 
9.90648 
9-90639 
9.90630 
9  .  90  620 

9 
9 
9 

10 

45 
44 
43 
42 
41 

.1 

.2 

•3 
•4 

•I 

I 
>, 

i 

c 

.8 
1-6 

!-4 
r.2 
).o 

0 

20 

21 
22 
23 
24 

9.77268 
9.77285 
9.77302 
9-773I9 
9-77336 

17 
»7 
17 
»7 
17 

9.86656 
9.86683 
9.86709 
9-86  736 
9.86  762 

27 
26 
27 

26 

0.13344 
0.13317 
0.13291 
0.13264 
0.13  238 

9.90611 
9.90602 
9.90592 
9-90583 
9-90574 

9 

9 
xo 

9 
9 

~w 

fs 

11 

.0 

:l 

•9 

K 

Ii 
1^ 

I( 

).o 

j.6 

M 

)  2 

2 
3 

29 

9-77353 
9-77370 
9-77387 
9.77405 
9.77422 

»7 
17 
18 

17 
17 

9.86789 
9.86815 
9.86842 
9.86868 
9.86894 

26 
27 
26 
26 

0.13  211 
0.13  185 
0.13  158 
O.I3I32 

0.13  106 

9-90565 
9.90555 
9.90546 
9-90537 
9.90527 

9 

10 

9 
9 
xo 

35 
34 

33 
32 
3i 

.1 

.2 
•3 

] 

17 

[-7 
5-4 

ij 

30 

3i 
32 

!  33 

1  34 

9-77439 
9-77456 
9-77473 
9.77490 

9-77507 

»7 

»7 
»7 
'7 

17 

9.86921 
9.86947 
9.86974 
9.87000 
9.87027 

26 
27 
26 
27 
26 

0.13079 
0.13053 
0.13026 
0.13000 
0.12973 

9.90518 
9.90509 

9-90499 
9.90490 
9.90480 

9 

9 
10 

9 
xo 

80 

1 

:! 

.9 

! 

1C 

i 

«; 

i 

J-5 

5.2 

'i 

j.6 
53 

!$ 
12 

39 

9-77524 
9-77541 
9-77558 
9-77575 
9-77592 

'7 
»7 
»7 
17 
17 

9-87053 
9.87079 
9.87  106 

9-87  132 
9.87  158 

26 
27 
26 
26 

0.12947 

0.12  921 
0.12894 
0.12868 
O.I2  842 

9.90471 
9.90462 
9.90452 
9-90443 
9-90434 

9 

9 
xo 

9 
9 

25 
24 
23 

22 
21 

.1 

.2 

16 

[.6 

i.  2 

40 

4i 
42 

43 

44 

9  .  77  609 
9.77626 

9-77643 
9  77660 
9.77677 

17 
17 
i? 
17 
17 

9-87  185 
9.87211 
9.87238 
9.87264 
9.87290 

26 
27 
26 
26 

0.12  815 
O.I2  789 
O.I2  762 
0.12  736 
0.12  710 

9.90424 
9.90415 
9.90405 
9.90396 
9.90386 

9 

xo 

9 
xo 

20 

19 

il 

•3 

•4 

:I 

.7 

* 
i 

( 
f 

3 
;:o- 

>.6 

1.2 

45 
46 

44I 
49 

9.77694 
9.77711 
9.77728 

9-77  744 
9.77761 

17 
17 
16 

17 

9.87317 

9-87343 
9.87369 
9.87396 
9.87422 

26 
26 
27 
26 
26 

0.12  683 
0.12  657 
0.12  631 
0.12  604 
0.12  578 

9.90377 
9.90368 
9-90358 
9-90349 
9-90339 

9 

9 
xo 

9 
xo 

15 

14 
13 

12 
II 

.8 
9 

i 

u 
i< 

0 

2.8 

J-4 

9 

50 

5i 

52 
53 
54 

9.77778 

9-77795 
9.77812 
9-77829 
9.77846 

J7 
»7 
17 
'7 

16 

9.87448 

9.87475 
9-87501 
9.87527 
9.87554 

27 
26 
26 
27 
26 

0.12552 
O.J2525 
0.12499 
0.12473 
0.12446 

9-90330 
9.90320 
9.90311 
9.90301 
9.90292 

9 
xo 

9 
xo 

9 

10 

I 

.1      i 

.2        2 

•3     3 
•4     A 

'?      f 

.0 
.0 
.0 
.0 

.0 

?:2 

ai 

4-5 

11 

11 

59 

9.77862 
9.77879 
9.77896 

9.779I3 
9.77930 

'7 
17 
'7 
'7 

16 

9.87580 
9.87606 
9-87633 
9-87659 
9.87685 

26 
2? 
26 
26 
26 

O.  12  42O 
0.12394 
0.12367 
O.I234I 
O.I23I5 

9.90  282 
9.90273 
9  .  90  263 
9.90254 
9.90244 

9 
xo 

9 
xo 

5 
4 
3 

2 
I 

.6     c 

:l  I 

•9     9 

.0 
.0 
.0 
.0 

1.1 

C 

00 

9.77946 

9.87711 

0.12  289 

9-90235 

9 

0 

L.  Cos. 

d. 

L.  Cotg. 

c.d. 

L.  Tang. 

L.  Sin. 

d. 

9 

Pro] 

).  . 

Pts. 

53° 

LOGARITHMS  OF  SINE,  COSINE,  TANGENT  AND  COTANGENT,  ETC.      67 


1                                       37° 

t 

1»  Sin. 

d. 

L.  Tang1. 

c.d. 

L.  Cotg-. 

L.  Cos. 

d. 

6JT 

Prop.  Pts. 

0 

I 

2 

3 

4 

9.77946 
9.77963 
9.77980 

9-77997 
9.78013 

17 
17 

16 

'7 

16 
17 
»7 
16 

»7 

17 
»7 
16 

16 
«7 
17 
16 

«7 

16 

17 
16 

17 
16 

16 
17 
16 
17 
16 
16 
»7 
16 
»7 

16 
i7 
16 

16 

16 
17 
16 
16 
16 

16 

16 

«7 

16 
16 
16 

16 
17 
16 
16 
16 

9.87711 
9-87738 
9-87764 
9.87  790 

9-87817 

27 
26 
26 
27 
26 
26 
26 
27 
26 
26 

0.12  289 
0.12  262 
0.12  236 
0.12  210 
0.12  183 

9-90235 
9.90225 
9.90  216 
9.90206 
9.90197 

xo 

9 
xo 

9 
xo 

9 
xo 

9 
xo 
xo 

9 
xo 

9 
xo 
xo 

9 
xo 

9 
xo 
xo 

9 
xo 
xo 

9 

10 

xo 

9 
xo 
xo 
9 
xo 
xo 

9 
xo 

10 

xo 

.9 

10 

xo 
xo 

9 
xo 

10 

xo 
9 
xo 

IO 

xo 
xo 
9 
xo 

10 

xo 
xo 
xo 

9 
xo 
xo 

10 

xo 

.1 

.2 

•3 
•4 

i 
:l 

-9 
.1 

.2 

•3 

•4 

4 

.1 

.2 

•3 
•4 

:I 

•9 

.2 

•  3 

•4 

'.S 
•  9 

.1      l 

.2       i 

•3    : 

•4      4 

V 

•91    * 

«7 

2.7 

H 

10.8 

'3-5 
16.2 
18.9 

21.6 

24.3 

2.6 

5-2 

7.8 

10.4 
13.0 
15.6 
18.2 

20.8 

23  4 

17 

1-7 

3-4 
5.1 
6.8 
8-5 

10.2 

"•I 

13-6 
15-3 

16 
1.6 

|! 

8.0 
9-6 

II.  2 

12.8 

14.4 

to         g 

.0    0.9 

5.0       1.8 

5-0     2.7 
[.o     3.6 
>.o     4-5 
>-o     5.41 
r.o     6.3 
5.0.    7.2 
i.o[    8.1 

I 
I 

9 
10 

12 
13 

9.78030 

9.78047 
9-78063 
9.78080 
9.78097 

9.87843 
9.87869 
9.87895 
9.87922 
9.87948 

O.I2I57 
0.12  131 
0.12  ID? 
0.12078 
0.12052 

9.90187 
9.90  178 
9.90  168 
9-90  159 
9-90  149 

55 
54 
53 
52 

9.78113 
9-78130 
9.78147 
9-78163 
9.78  180 

9.87974 
9.88000 
9.88027 
9-88053 
9.88079 

26 
27 

26 
26 

0.12026 
0.12000 

O.II973 
O.II947 

o.ii  921 

9.90139 
9-90  13° 

9.90  120 
9.90  III 

9.90  ioi 

50 

42 
48 

~iT 

44 
43 
42 
41 

!i 
\l 

19 
"20" 

21 
22 

23 

24 

9.78197 
9-78213 
9.78230 
9.78246 
9.78263 

9.88  105 
9  .88  131 
9.88  158 
9.88  184 
9.88210 

26 

87 

26 
26 
26 
26 
27 
26 
26 
26 
26 
27 

26 
26 
26 
26 
26 
27 
26 
26 
26 
26 
26 
26 
26 
27 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
27 
26 
26 

26 
26 
26 
26 
26 

o.ii  895 

O.II  869 

o.ii  842 
o.ii  816 
o.ii  790 

9.90091 

9.90  082 

9.90072 
9-90063 
9-90053 

9.78280 
9.78296 

9-78313 
9.78329 
9.78346 

9.88236 
9.88262 
9.88289 
9.88315 
9.88341 

o.ii  764 
o.ii  738 
o.ii  711 
o.ii  685 
o.ii  659 

9.90043 
9.90034 

9  .  90  024 
9.90014 
9.90005 

1 

27 
28 
29 

9.78362 
9.78379 
9-78395 
9.78412 
9.78428 

9.88367 

9.88393 
9.88420 
9.88446 
9.88472 

o.ii  633 
o.ii  607 
o.ii  580 
o.ii  554 
o.ii  528 

9-89995 
9.89  985 
9.89976 
9.89966 
9.89956 

35 
34 
33 
32 
3' 

30 

32 

33 

34 

9.78445 
9.78461 
9.78478 

9.78494 
9.78510 

9.88498 
9.88524 
9-88550 
9.88577 
9.88603 

o.ii  502 
o.ii  476 
o.ii  450 
o.ii  423 

0.11397 

9.89947 
9-89937 

9'.899i8 
9.89908 

30 

29 
28 

% 

I 

39 

41 
42 
43 

44 

9.78527 

9.78543 
9-78560 

9-78592 

9.88629 
9-88655 
9.88681 
9.88707 
9-88  733 

o.ii  371 
o.ii  345 
o.ii  319 
o.ii  293 
o.ii  267 

9.89898 
9.89888 
9.89879 
9.89869 
9-89859 

25 
24 

23 

22 
21 

"20" 

|2 

9.78609 
9-78625 
9  .  78  642 
9.78658 
9-78674 

9-88  759 
9.88  786 
9.88812 
9.88838 
9.88864 

o.ii  241 
o.ii  214 
o.ii  188 
o.ii  162 
o.ii  136 

9.89849 
9.89840 
9.89830 
9  .  89  820 
9.89810 

47 
48 

49 

9.78691 
9.78  707 
9.78723 

9.88.890 
9.88916 
9.88942 
9.88968 
9.88994 

O.II  IIO 

o.ii  084 
o.ii  058 
o.ii  032 

O.II  006 

9.89801 
9.89  791 
9.89  781 
9.89  771 
9.89  761 

15 
14 
13 

12 
II 

lo~ 

I 
I 

50 

52 
53 
54 

9.78772 
9.78788 
9-78805 
9.78821 
9.78837 

9.89020 
9.89046 
9.89073 
9.89099 
9-89  125 

0.10980 
0.10954 
o.io  927 
0.10901 
0.10875 

9.89  752 
9.89  742 
9.89732 
9.89  722 
9.89  712 

1 

58 
59 

9-78853 
9.78869 
9.78886 
9.78902 
9.78918 

9.89  151 
9.89  177 
9.89203 
9  .  89  229 
9-8925? 

0.10849 
0.10823 

0.10797 

o.io  771 

0.10745 

9.89  702 
9.89693 
9.89683 
9.89673 
9-89663 

5 
4 
3 

2 

I 

nr 

GO 

9-78934 

9.89281 

o.io  719 

9  89653 

L.  Cos. 

d. 

L.  Cotfir. 

c.d. 

L.  Tang-. 

L.  Sin. 

d. 

t 

Prop.  Pts. 

52° 

68 


TABLE  IV. 


38° 

1 

L.  Sin. 

d. 

L.  Tanar. 

c.d. 

L.  Cotg. 

L.  Cos. 

d. 

r 

ro] 

). 

Pts. 

0 

2 

3 
4 

9-7*934 
9.78950 
9.78967 
9.78983 
9.78999 

16 

17 

16 
16 
16 

9.89281 
9-89307 
9.89333 
9-89359 
9-89385 

26 
26 
26 
26 
26 

o.io  719 
0.10693 
0.10667 
0.10641 
0.10615 

9-89653 
9.89643 

9-89633 
9  .  89  624 
9.89  614 

xo 
xo 
xo 
xc 
xo 

00 

9 

11 

.1 

2 

2 

2 
e 

6 
.6 

2 

25 

2-5 
e  o  1 

I 
i 

9 

9.79015 
9.79031 
9.79047 
9.79063 
9.79079 

16 
*6 
16 
16 

16 

9.89411 

9.89437 
9.89463 

9.89489 
9-89515 

76 
26 
26 
26 
26 

0.10589 
o.io  563 

0.10537 

o.io  511 
0.10485 

9.89604 
9.89594 
9-89584 
9-89574 
9.89564 

xo 
xo 
xo 
xo 

55 
54 
53 

52 
51 

.3 

:S 

.7 

7 

10 

13 

;i 

.8 
-4 

.0 

.6 

.2 

75 

10.  0 

12.5 
15.0 

^•S  i 

10 

ii 

12 
13 
H 

9-79°95 
9.79111 
9.79128 
9.79  144 
9.79160 

16 

17 
16 
16 
16 

9.89541 
9.89567 

9.89593 
9.89619 
9.89645 

26 
26 
26 
26 
26 

0.10459 
0.10433 

0.10407 
0.10381 

0.10355 

9-89554 
9.89544 
9.89534 
9.89524 
9.89514 

xo 
xo 
xo 
xo 

50 

3 

8 

.8 
•9 

20 
23 

.8 
•4 

i 

20.  o 
22.5 

7 

it 

\l 

19 

9.79176 
9.79192 
9  .  79  208 
9.79224 
9.79240 

16 
16 
16 
16 

16 

9.89671 
9.89697 
9.89723 
9.89749 
9.89775 

26 
26 
26 
26 
26 

0.10329 
0.10303 

0.10277 
0.10251 
0.10225 

9.89504 
9-89495 
9.89485 
9-89475 
9.89465 

9 
xo 
xo 
xo 

45 
44 
43 

42 
41 

.1 
.2 

•3 

•  4 

•5 

i 

i 

! 

-7 

1-4 

si 

».s 

"20 

21 
22 

23 
24 

9.79256 
9.79272 
9.79288 
9-79304 
9.79319 

16 
x6 
x6 
IS 

16 

9.89801 
9.89827 
9.89853 
9-89879 
9.89905 

26 

26 
26 
26 
26 

o.io  199 
o.io  173 
o.io  147 

O.IO  121 

0.10095 

9-89455 
9-89445 
9-89435 
9.89425 
9.89415 

xo 
xo 
xo 
xo 

40 

It 

9 

:l 

-9 

I 

i; 

i. 

[-9 
*-6 

5-3 

3 

3 

29 

9-79335 
9-79351 
9.79367 

9-79383 
9-79399 

16 
16 
16 
16 
x6 

9.89931 

9.89957 
9,89983 
9.90009 
9-90035 

26 
26 
26 
26 
26 

0.10069 
0.10043 
0.10017 
0.09991 
0.09965 

9.89405 
9.89395 
9-89385 
9.89375 
9.89364 

IO 

xo 
xo 

XI 

35 
34 
33 

32 
31 

.1 

.2 

•3 

.4 

i 
i 

i 

^ 
f 

6 
.6 

3 

>  4 

15 
'5 
3-o 

*j 

30 

3i 
32 
33 

34 

9-794I5 
9-79431 
9-79447 
9-79463 
9.79478 

16 
16 
16 
15 
16 

9.90061 

9  .  90  086 

9.90  112 

9.90138 
9.90  164 

25 
26 
26 
26 
26 

0.09  939 
0.09  914 
0.09  888 
0.09  862 
0.09  836 

9.89354 
9.89344 
9.89334 
9.89324 
9.89314 

xo 

10 

xo 

IO 

30 

2! 

11 

1 
i 

.9 

i 
c 
ii 

12 
It 

1.0 

).6 

.2 
5.8 

I--4 

7-5 
9.0 
10.5 

12.  0 

'3  5 

9 
9 

39 

9-79494 
9.79510 
9.79526 
9-79542 
9-79558 

16 
16 
16 
16 

9.90190 

9.90216 
9.90242 
9.90268 
9.90294 

26 
26 
26 
26 
26 

0.09810 
0.09  784 
0.09  758 
0.09  732 
0.09  706 

9.89304 
9.89294 
9.89284 
9.89274 
9.89  264 

xo 
xo 

10 

xo 

25 
24 
23 

22 
21 

.1 

.2 

ii 

[.i 

2.2 

40 

4i 
42 

43 
44 

9-79573 
9-79589 
9-79605 
9.79  621 
9.79636 

x6 
16 
16 
IS 

16 

9.90320 
9.90346 
9.90371 
9.90397 
9.90423 

26 
25 
26 
26 
26 

0.09  680 
0.09  654 
0.09  629 
o  .  09  603 
0.09577 

9.89254 
9.89244 

9-89233 
9.89223 
9.89213 

xo 

XX 

xo 
xo 

20 

ii 

•3 
•4 

•7 

i 

( 

3-3 
[>4 

7-7 

2 

;*7 

49 

9.79652 
9  79668 
9.79684 
9.79699 
9-797I5 

16 

16 

IS 

16 

16 

9.90449 

9-90475 
9-90501 
9.90527 

9-90553 

26 
26 
26 
26 

0.09551 
0.09525 
0.09499 
0.09473 
0.09447 

9.89  203 

9-89193 
9.89  183 

9-89  173 

9.89  W2 

zo 
xo 
xo 

XX 

15 
14 
13 

12 

II 

.8 
•9 

3 

< 

i 

0 

3.8 
)-9 

9 

50 
5i 
52 
53 

54 

9-79  731 
9.79746 

9-79  762 
9.79778 
9  79  793 

IS 
16 
16 

IS 

16 

9.90578 
9.90604 
9.90630 
9.90656 
9  .  90  682 

26 
26 
26 
26 
26 

0.09  422 
0.09  396 
0.09370 
0.09344 
0.09318 

9.89I52 
9.89142 
9.89  132 
9.89  122 
9.89  112 

xo 
xo 
xo 
xo 

10 

I 
I 

.1 

.2 

•3 
•  4 

] 
2 

3 

A 

I 

.0 

.0 

-o 

.0 

.0 

0.9 

ii 

4-5 

55 
56 

fi 

59 

9.79809 
9-79825 
9.79840 
9.79856 
9.79872 

x6 
15 
x6 
16 

9  .  90  708 
9-90734 
9-90759 
9.90  785 
9.90  811 

26 
25 
26 
26 
26 

0.09  292 
0.09  266 
0.09  241 
0.09  215 
0.09  189 

9.89  101 

9.89  091 
9.89081 
9.89071 

9  .  89  060 

xo 

10 

xo 

IX 

5 
4 
3 

2 

I 

:1 

.9 

I 

9 

.0 
.0 
.0 

n 

K 

60 

9.79887 

9.90837 

0.09  163 

9  .  89  050 

0 

L.  €os. 

d. 

L.  Cotg. 

c.d. 

L.  Tang. 

L.  Sin. 

d. 

/ 

p 

roj 

). 

PtSr 

51° 

LOGARITHMS  OF  SINE,  COSINE,  TANGENT  AND  COTANGENT,  ETC.     69 


39° 

t 

L.  Sin. 

d. 

L.  Tang. 

c.d. 

L.  Cotg. 

L.  Cos. 

d. 

Proj 

.1 

»ts. 

0 
I 

2 

3 
4 

9.79887 

9-79903 
9.79918 

9-79934 
9-79950 

16 

15 
16 
16 
15 

9.90837 
9.90863 
9.90889 
9.90914 
9-90940 

26 
26 

as 
26 
26 

0.09  163 
0.09  137 
0.09  in 
0.09  086 
0.09060 

9  .  89  050 
9  .  89  040 
9.89030 
9.89020 
9.89009 

xo 

IO 
10 

II 

IO 

00 

ii 

11 

.1 

.2 

21 
2 

5 

5 

.6 

.2 

\ 

I 

9 

9.79981 
9.79996 
9.80012 
9.80027 

16 
IS 
16 
15 

16 

9.90966 
9.90992 
9.91  018 
9.91  043 
9.91069 

26 
26 
as 
26 
26 

0.09034 
0.09  008 
0.08982 
0.08957 
0.08931 

9.88999 
9.88989 
9.88978 
9.88968 
9.88958 

10 
XI 

xo 
xo 
xo 

55 
54 
53 

52 
51 

•3 
•4 

'•7 

7 

10 

13 

!i 

.8 

•4 

.0 

.6 

.2 

10" 
ii 

12 

'3 

14 

9.80043 
9.80058 
9  .  80  074 
9.80089 
9.80  105 

IS 

16 

IS 
16 

15 

9.91095 

9.91  121 
9.91  147 
9.91  172 
9.91   198 

26 
26 

26 
26 

0.08905 
0.08879 
0.08853 
0.08828 
0.08802 

9.88948 

9-88937 
9.88927 

9.88917 
9.88906 

XX 

xo 

IO 
XX 

xo 

50 

4i 
48 

.8 
•9 

20 

23 

a 

.8 
•4 

5 

19 

9.80  120 
9.80  136 
9.80151 
9.80  166 
9.80  182 

16 
IS 
IS 

16 

15 

9.91  224 

9  9i  250 
9.91  276 

9-91  301 
9.91  327 

26 
26 

as 
26 
26 

0.08  776 
0.08  750 
0.08  724 
0.08  699 
0.08673 

9  .  88  896 
9.88886 
9.88875 
9.88865 
9.88855 

IO 
XX 

xo 
xo 

jj 

45 
44 
43 

42 
41 

.1 

.2 

•3 

•4 

2 

5 

7 

10 
12 

•5 
.0 

•  5 
.0 

•5 

20 

21 
22 
23 
24 

9.80  197 
9.80213 
9.80228 
9.80244 
9-80259 

16 

IS 
16 

9  9i  353 
9-91  379 
9.91404 
9.91430 
9.91  456 

26 
85 
26 
26 
26 

0.08  647 
0.08  621 
0.08  596 
0.08  570 
0.08544 

9.88844 
9-88834 
9.88824 
9.88813 
9.88803 

xo 

IO 
XX 
IO 

40 

3 

11 

9 

J5 
I? 
2C 
22 

•5 
.0 

-5 

29 

9.80274 
9  .  80  290 
9-80305 
9.80320 
9-86336 

x6 
15 
IS 

x6 

9.91  482 
9.91  507 
9  9i  533 
9-91  559 
9  9i  585 

as 
26 
26 
26 

0.08518 
0.08493 
0.08467 
0.08441 
0.08  415 

9.88793 
9.88782 
9.88772 
9.88  761 
9.88751 

XX 

IO 
IX 

xo 

35 
34 
33 

32 
31 

.1 

.2 

•3 

A 

1 
1 

3 

i 

6 
.6 

1:5 

30 

32 
33 
34 

980351 

9.80366 
9.80382 
9.80397 
9.80412 

IS 

16 

IS 
IS 
16 

9.91  610 
9.91  636 
9.91  662 
9.91  688 
9.91  713 

26 
26 
26 
as 
26 

0.08  390 
0.08  364 
0.08  338 
0.08  312 
0.08287 

9.88741 
9  88730 
9.88  720 
9.88709 
9.88699 

IX 
IO 
IX 
IO 

30 

i 

•I 

.9 

J 

< 
i] 
i: 
K 

!-4 
\l 

.2 

1.8 

L-4 

39 

9.80428 
9.80443 
9-80458 
9.80473 
9.80489 

IS 

15 
IS 

16 

9-9i  739 
9.91  765 

9  9i  791 
9.91  816 
9  91  842 

26 
26 
as 
26 
26 

0.08  261 
0.08235 
0.08  209 
0.08  184 
0.08  158 

9.88688 
9.88678 
9.88668 
9.88657 
9.88647 

xo 
xo 
II 

IO 

25 
24 
23 

22 
21 

.1 

.2 

] 

<5 
5 

4i 
42 

43 

44 

9  .  80  504 
9  80519 
980534 
9  80550 
9  80565 

IS 

16 
IS 

9  91  868 
9  91  893 
9.91  919 

9-9»  945 
9.91971 

as 

26 

26 
26 

0.08  132 
0.08  107 
0.08081 
0.08055 
0.08  029 

9.88636 
9  .  88  626 
9  88615 
9.88605 
9.88594 

IO 

II 

10 

II 

20 

il 

3 
•  4 

7 

i 
( 

( 

1C 

kS 

)  O 

r.5 

)  o 
>-5 

s 

49 

9  .  80  580 
9  80  595 
9  80  610 
9  80  625 
9  80  641 

15 

'5 

16 

9.91  996 

9   92  022 

9  92  048 
9  92073 
9.92099 

26 
26 
25 
26 
26 

0.08004 
o  07  978 
o  07952 

0.07927 

0.07  901 

9  88  584 
988573 
9.88563 
9  88552 
9-88542 

II 

IO 

II 

IO 

15 
14 

13 

12 

II 

.8 
•9 

i: 

i; 

d 

s.o 
J  5 

xo 

50 

52 
53 
54 

9  80  656 
9.80  671 
9.80686 
9.80  701 
9.80  716 

IS 

15 

IS 

15 

9.92  125 
9-92150 
9.92176 

9  .  92  202 
9.92227 

85 

26 
26 

as 
06 

0.07875 

0.07  850 
0.07  824 
0.07  798 
0.07  773 

9.88531 
9.88521 
9.88510 

9-88499 
9.88489 

xo 

II 
II 

10 

10 

I 
I 

.1     1 
.2     : 

.3    : 

.4    / 

•5     l 

t.i 

1.2 

5-3 
t-4 

i-j 

I.O 

2.O 

3-o 
4.0 

i-° 

55 
56 

59 

9  80731 
9  80  746 
9  80  762 
9.80777 
9  .  80  792 

15 

16 
IS 

9.92279 
9.92304 
9.92330 
9.92356 

26 
85 
26 
26 

0.07  747 
0.07  721 
0.07696 
0.07  670 
0.07  644 

Q  88  478 
9  .  88  468 
9.88457 
9.88447 
9.88436 

10 
XI 
10 
XX 

5 
4 
3 

2 

I 

.6     ( 

:!  i 

•9     S 

>.t> 

7-7 
5.8 

>-9 

6.0 
7.0 
8.0 
9.0 

(iO 

9.80807 

9.92381 

2S 

0.07  619 

9.88425 

0 

L.  Cos. 

d. 

L.  Cotg. 

c.d. 

L.  Tang. 

L.  Sin. 

d. 

t 

Pro] 

P.: 

Pts. 

50° 

TABLE  IV. 


40° 

t 

L.  Sin. 

d. 

L.  Tang. 

c.d. 

L.  Cotg. 

L.  Cos. 

d. 

Prop.  Pte. 

0 

9.80807 

9.92381 

26 

0.07619 

9.88425 

60 

i 

9.80822 

9.92407 

0.07  593 

9.88415 

59 

2 

9.80837 

is 

9-92433 

0.07567 

9.88404 

58 

36 

3 

9.80852 

15 

9.92458 

25 
26 

0.07  542 

9.88394 

10 

57 

j 

2  6 

4 

9.80867 

9.92484 

26 

0.07  516 

9.88383 

M 

56 

.2 

e.2 

1 

9.80882 
9.80897 
9.80912 
9.80927 

15 
15 
15 

9.92  510 
9-92535 
9-92561 
9.92587 

25 
26 
26 

0.07490 
0.07465 
0.07439 
0.07413 

9.88372 
9.88362 

9-88351 
9.88340 

10 

II 
II 

55 
54 
53 

S2 

•3 
•4 

7-8 
10.4 
130 
15.6 

9 

9.80942 

15 

9.92  612 

25 
26 

0.07388 

9.88330 

II 

•  7 

lS.2 

ii 

12 

9.80957 
9.80972 
9.80987 

15 

IS 

9.92  638 
9.92663 
9.92  689 

25 

26 

0.07362 
0.07337 
0.07311 

9.88319 
9.88308 
9.88298 

II 

IO 

50 

49 
48 

.8 
•9 

20.8 

23-4 

13 

9.8l  002 

15 

9.92  715 

20 

0.07285 

9.88287 

47 

14 

9.8l  017 

15 
15 

9-92  740 

25 
26 

0.07260 

9.88276 

IO 

46 

»5 

!i 

9.8l  032 
9.8l  047 

IS 

9.92  766 
9.92  792 

26 

0.07  234 
0.07  208 

9.88266 
9.88255 

XX 

45 
44 

.1 

.2 

2.5 

5-0 

IS 

9.81  061 
9.81  076 

14 
IS 

9.92817 
9.92  843 

25 
26 

0.07  183 
0.07  157 

9.88244 
9-88234 

to 

43 
42 

•3 
•4 

7-5 

IO.O 

19 

9.81  091 

IS 
15 

9.92868 

25 
26 

0.07  132 

9.88223 

I] 

41 

•5 

12.5 

20 

21 

9.81  106 
9.81  121 

IS 

9.92894 
9.92  920 

26 

0.07  106 
0.07080 

9.88212 
9.88201 

XI 

*9 

i 

17-5 
20  o 

22 

9.81  136 

*5 

9  92945 

25 

?f> 

0.07055 

9.88  191 

38 

o 

22.  C 

23 

24 

9.81  151 
9.81  166 

IS 

9.92971 
9.02  996 

25 
26 

0.07029 
o  .  07  004 

9.88  180 
9.88  169 

II 

11 

11 

9.81  180 
9.81  195 

9.93022 
9.93048 

26 

0.06  978 
0.06952 

9.88  158 
9  .  88  148 

IO 

35 
34 

*    f 

11 

9.8l  210 
9.81  225 

15 
IS 

9-93073 
9.93099 

25 
26 

0.06927 
0.06901 

9.88137 
9.88126 

xr 

33 
32 

.2 

••J 

3-0 

29 

9.8l  240 

15 

9  93  124 

25 
26 

0.06876 

9.88  115 

'. 
.4 

60 

30 

3' 
32 

9.8l  254 
9.8l  269 
9.8l  284 

15 

IS 

9-93  ISO 
9  93  175 

9.93  2OI 

25 
26 

0.06  850 
0.06  825 
0.06  799 

9.88105 
9.88094 
9.88083 

ii 

IX 

30 

3 

i 

7-5 
9.0 
10.5 

33 
34 

9.8l  299 
9.8l  314 

15  . 
15 

9-93227 
9.93252 

26 
25 
26 

0.06  773 
0.06  748 

9.88072 
9.88061 

IX 

10 

2 

•9 

12.0 
135 

P 
8 

9.81328 

9-8i  343 
9-8i  358 
9.81  372 

15 

15 
14 

9.93278 
9-93303 
9-93329 

9-93354 

25 
26 
25 

0.06  722 
0.06  697 
0.06  671 
0.06  646 

9.88051 
9.88040 
9  88  029 
9.88018 

XI 

II 
II 

25 
24 
23 

22 

N 

I   A. 

39 

9.81387 

15 

9.9338o 

26 

0.06  620 

9.88007 

21 

.2 

I'i 

40 

9.81  402 

9.93406 

0.06  594 

9.87996 

20 

•  3 

4-2 

41 

9.81417 

15 

9-93431 

25 

0.06  569 

9.87985 

19 

.4 

5-6 

42 

9.81  431 

14 

9  93457 

26 

0.06  543 

9  87975 

18 

7.0 

43 
44 

9.81446 
9.81  461 

15 
15 

9.93482 
9-93  5o8 

25 
26 

0.06  518 
0.06492 

9.87964 
9.87953 

II 

11 

•7 

8.4 
9.8 

45 

9  81475 

9-93533 

25 

0.06467 

9.87942 

IS 

.8 

II.  2 

46 

9.81  490 
9-8i  505 

IS 

15 

9-93  559 
9-93  584 

26 
25 

0.06441 
0.06416 

9.87931 
9.87920 

II 

14 
13 

9 

12.6 

48 

9.81  519 

14 

9.93610 

26 

0.06390 

9.87909 

12 

49 

9  81  534 

IS 

9.93636 

26 

0.06  364 

9.87898 

II 

IX           10 

50 
52 

9.81  549 
9-81563 
9.81  578 

14 
15 

9.93661 
9-93687 
9  93  712 

25 
26 
25 

0.06339 
0.06313 
0.06  288 

9.87887 
9.87877 
9.87866 

xo 
II 

10 

.1        I.I       I.O 
.2       2.2      2.0 

•3     33     3-0 

53 

9.81  592 

14 

9-93  73s 

26 

0.06  262 

9-87855 

IX 

7 

.4    4.4   4.0 

54 

9.81  607 

15 

9-93  763 

25 

0.06237 

9.87844 

6 

-I  n  I'i 

1 

9.81  622 
9.81  636 
9.81  651 
9.81  665 

14 

IS 

14 

9  93  789 
9-93  814 
9.93840 

25 
26 
25 

0.06211 
0.06  186 
0.06  160 
0.06  135 

9-87833 
9.87822 
9.87811 
9.87800 

XX 
XX 

XX 

5 
4 
3 

2 

.6     6.6     6.0 

•1  11  I:S 

9     9.9    9.0 

59 

9.81  680 

15 

9.93891 

26 

0.06  109 

9.87789 

XX 

I 

GO 

9  81  694 

9.93916 

25 

0.06084 

9.87778 

0 

L.  Cos. 

d. 

L.  Cotg. 

c.d. 

L.  Tang. 

L.  Sin. 

d. 

t 

Prop.  Pts. 

49°                                         | 

LOGARITHMS  OF  SINE,  COSINE,  TANGENT  AND  COTANGENT,  ETC.     7, 


1 

41° 

, 

L.  Sin. 

d. 

L.  Tang. 

c.d. 

L.  Cotg. 

L.  Cos.  ' 

d. 

Proj 

).  Pts. 

ho" 

2 

* 

9.81  694 
9.81  709 
9.81  723 
9.81  738 
9.81  752 

«s 

14 
15 
14 

15 

9.93916 
9-93942 
9-93967 
9-93  993 
9.94018 

26 
25 
26 

25 
26 

0.06084 
0.06058 
0.06  033 
0.06007 
0.05  982 

9.87778 
9.87  767 
9.87  756 
9-87  745 
9-87  734 

IX 

zz 

ZI 
IZ 
ZI 

60 

9 
9 

.1 

.2 

a6 
2.6 

C.2 

* 

9 

9  81  767 
9.81  781 
9  81  796 
9.81  810 
9.81825 

14 
15 
14 

15 

14 

9.94044 
9.94069 

9-94095 
9.94  120 

9-94  H6 

25 
26 
25 
26 
25 

0.05  956 
0.05  931 
0.05  905 
0.05  880 
0.05  854 

9.87723 
9  87  712 
9.87  701 
9.87690 
9.87679 

zz 
zz 
zz 
zz 
JI 

55 
54 
53 

52 
51 

•3 

:! 

.7 

7-8 
10.4 
13.0 
15.6 
18.2 

10" 
ii 

1    I2 
*3 
14 

9.81  839 
9-81854 
9.81868 
9.81  882 
9.81897 

IS 
14 
14 
15 
14 

9.94171 

9-94  197 
9.94222 
9.94248 
9-94273 

26 

25 
26 
25 
26 

0.05  829 
0.05  803 
0.05  778 
0.05  752 
0.05  727 

9.87  668 
9.87657 
9.87646 

9.f7635 
9.87624 

zz 
zz 
zz 
zz 
zz 

50 

9. 
9. 

.8 
•9 

20.8 

23.4 

m 

15 
10 

17 
18 
19 

9.81  911 
9.81  926 
9.81  940 

9  81955 
9.81  969 

»S 
H 

15 
14 

14 

9.94299 
9-94324 
9-94350 
9  94375 
9,94401 

«S 
26 
25 
26 

25 

0.05  701 
0.05  676 
0.05  650 
0.05  625 
0.05  599 

9-87613 
9.87601 
9.87590 
9  87579 
,9-87568 

12 

zz 
zz 

ZI 

II 

45 
44 
43 

42 
41 

.2 
•3 

•4 
'I 

2-5 

5-o 
7-5 

IO.O 

12  5 

1C    O 

20 

21 

22 

1   23 

24 

9  81  983 
9.81  998 

9.82  OI2 

9  .  82  026 
9.82  041 

'5 
M 
14 
»S 
14 

9  94  426 
9  94452 
9-94477 
9  94503 
9.94528 

26 
25 
26 
25 
26 

0.05  574 
0.05  548 
0.05  523 
0.05497 
0.05472 

9.87557 
9.87546 

9  87535 
9.87524 

9  87513 

ZI 

zz 
zz 

ZI 

ia 

40 

37 

36 

i 

•9 

»5-° 
17-5 
20.  o 

22.5 

hs~ 

26 

h 

\   29 

9.82055 
9  82  069 
9.82084 
9.82098 

9.82  112 

14 
IS 

14 
14 

9-94  554 
9-94579 
9.94604 
9  94630 
9  94655 

25 

25 
26 
25 
26 

0.05  446 
0.05421 
0.05  396 
0.05  370 
0-05  345 

9.87501 
9.87490 
9.87479 
9.87468 
9  87  457 

tz 
II 
II 

ZI 
ZI 

35 

34 
33 

32 
31 

i 

.2 

•3 
.4, 

«5 

"5 

30 

n 

30 

31 
I  32 

33 
34 

9.82  126 
9.82  141 
9-82  155 
9.82  169 
9.82  184 

IS 
14 
«4 
«S 

9.94681 
9.94706 
9-94732 
9-94757 
9-94  783 

25 

26 
25 
26 
25 

0.05319 
0.05  294 
0.05  268 
0.05  243 
0.05  217 

9  87446 
9  87434 
9.87423 
9  87412 
9.87401 

IS 

II 

ZI 

II 
zz 

30 

2! 

11 

ii 

:1 

•9 

7.5 
9.o 

lo-S 

12.  0 
'3-5 

P 
9 

39 

9.82  198 
9  82  212 

9  82  226 
9  .  82  240 
9  82255 

14 
M 
«4 
IS 

9.94808 
9-94  834 
9.948$9 
9.94884 
9.94910 

26 
25 
«S 
26 

o  05  192 
0.05  166 
o  05  141 
o  05  116 
o  05  090 

9  87390 
9.87378 
9  87  367 
9  87356 
9  87345 

13 

zz 
zz 

ZI 
ZI 

25 
24 
23 

22 
21 

.1 

.2 

14 

;i 

40 

41 
42 
43 
44 

9  82  269 
9  82283 
9  82297 
9.82  311 
9.82326 

14 
14 

14 
IS 

9  94935 
9.94961 
9  94986 
9.95012 
9  95037 

26 

25 
26 

25 

0.05  065 
005039 
0.05  014 
o  04988 
0.04963 

9  87334 
9.87322 
9  87311 
9.87300 
9.87288 

12 

II 
II 

12 
IZ 

20 

19 

il 

•3 
•4 

i 

•7 

g 

i;: 

9.8 

9 

9- 

49 

9  82  340 
9  82354 
9.82  368 
9.82382 
9  82396 

«4 
14 
14 
14 

9.95062 
995088 
9  95  »3 
9  95  '39 
9-95  164 

26 
25 

26 
25 

26 

o  04938 
0.04912 
0.04  887 

0.04  861 
o  04  836 

9  87277 
9  87266 
9  87255 
9  87  243 
9.87232 

zz 

IZ 
Z2 
ZI 

15 
14 
13 

12 
II 

.8 
•9 

i 

II.  2 
12.6 

a         II 

150 

51 
52 

53 

54 

9.82  410 
9  82424 
9.82439 
9  82453 
9.82467 

14 

»s 

14 

«4 

9-95  !90 
9-952I5 
9-95  240 
9-95  266 
9-95  291 

25 
as 

26 
«5 

26 

0.04  810 
0.04  785 
0.04  760 
0.04  734 
0.04  709 

9.87  221 
9.87209 
9.87  I98 
9.87  187 

9  87175 

la 
zz 
zz 
za 

ff 

10 

6 

.1      i 
.2     s 
•3     c 

•4      4 

.2       I.I 
(.4      2.2 

[6     3.3 

^8     4-4 

>0  ii 

56 

19 

59 

9.82481 
9.82495 
9-82509 
9-82523 
9  82537 

«4 
>4 
14 
«4 
14 

9-953I7 
9-95342 
9-95  368 
9-95393 
9.95418 

as 
26 
as 
«5 
96 

0.04683 
0.04  658 
0.04  632 
o  .  04  607 
0.04  582 

9.87  164 

9-87153 
9.87  141 

9-87  130 
9.87  119 

XI 

12 

xz 
11 

5 
4 

3 

2 
I 

Ii 

•9    « 

r.2     6.6 

U     7-7 
>  6     8.8 
>.8     9.9 

9-82551 

9-95444 

0.04  556 

9.87  107 

0 

1 

L.  Cos. 

d. 

L.  Cotgr. 

c.d. 

L.  Tang. 

L.  Sin. 

d. 

/ 

Pro] 

p.  Pts. 

4:8° 

2 

LAKLZ,  IV 

42° 

/ 

L.  Sin. 

d. 

L.  Tang. 

c.d. 

L.  Cotg. 

L.  Cos. 

d. 

Pro] 

).  ] 

Pts. 

0 

I 

2 

3 
4 

9.82551 
9.82565 
9.82579 
982593 
9.82  607 

14 
14 
14 
14 

9-95444 
9  95469 
9-95495 
9-95  520 
9-95  545 

25 
26 
25 
25 
26 

0.04556 
0.04531 
0.04505 
0.04480 
0.04455 

9.87  107 
9.87096 
9-87085 
9.87073 
9.87062 

ii 

IX 
12 
II 

60 

52 
58 

8 

2 

a 

2 

i 
.6 

2 

1 

1 

9 

9.82  621 
9-82635 
9  .  82  649 
9.82663 
9.82677 

«4 
14 
14 
»4 
14 

9-95571 
9-95  596 
9-95  622 
9-95  647 
9-95  672 

25 
26 
25 
25 
26 

o  04  429 
o  04  404 
0.04  378 

0.04353 

o  04328 

9.87  050 
9.87039 
9.87028 
9.87  016 
9  -  87  005 

II 
IX 
12 
II 
12 

55 
54 
53 
52 
5i 

•3 
.4 

:I 

.7 

7 

1C 

13 

!! 

.8 

•4 

.0 

.6 

.2 

10 

ii 

12 
13 
H 

9.82  691 
9.82  705 
9.82  719 
9-82  733 
9-82747 

H 
14 
J4 
»4 
14 

9.95698 
9-95  723 
9-95  748 
9-95  774 
9-95  799 

25 
25 
26 
25 
36 

o  .  04  302 

0.04277 
0.04252 
0.04  226 

0.04  201 

9.86993 
9  .  86  982 
9.86  970 
9.86959 
9.86947 

II 
l» 
IX 
12 

50 

3 

11 

.8 
•9 

2C 

23 

3 

.8 
•4 

5 

II 
\l 

19 

9.82  761 

9-82775 
9.82  788 
9.82802 
9.82816 

»4 

13 
14 
14 
1^ 

9.95  825 
9-95850 
9-95875 
9-95  901 
9  95  926 

25 
25 
26 
25 
26 

0.04  175 
0.04  150 
0.04  125 
0.04099 
0.04074 

9  .  86  936 
9  .  86  924 
9  86  913 
9.86902 
9.86890 

12 
XI 
II 

is 

ii 

45 
44 
43 
42 

41 

.1 

.2 

•3 
•4 

J 

2 

5 
5 

1C 
12 

•5 
.0 

'•5 
>.o 

••5 

20 

21 
22 

•23 
24 

9.82830 
9.82844 
9.82858 
9  .  82  872 
9.82885 

>4 
«4 
14 
»3 
14 

9-95  952 
9-95  977 
9.96002 
9  .  96  028 
9-96053 

25 
95 

•6 
*5 

o  04  048 
0.04023 
0.03998 
0.03  972 
0.03947 

9.86879 
9.86867 
9.86855 
9.86844 
9  .  86  832 

13 
12 
XX 
19 

40 

P 

H 

:i 

•9 

i; 

2C 
22 

•O 

r-5 

5   O 

'•5 

25 
26 

3 

29 

9.82899 
9.82913 
9.82927 
9.82941 
9  82955 

»4 
14 

14 
H 
13 

9.96078 
9.96  104 
9.96  129 

9.96  155 
9.96  180 

26 
25 
26 
25 

0.03  922 
0.03  896 
0.03871 
0.03  845 

o  03  820 

9.86821 
9.86  809 
9.86  798 
9.86786 
9-86775 

12 
IX 
12 
XX 

35 
34 
33 
32 
3i 

.1 

.2 

•3 

A 

1 
1 

t 

14 

1:1 
H 

80 

3i 
32 
33 
34 

9.82968 
9.82982 
9.82996 
9.83  oio 
9.83023 

14 
14 
14 
»3 
14 

9.96205 
9.96231 
9.96256 
9.96  281 
9.96  307 

26 
25 
25 
26 

0.03  795 
0.03  769 
0.03  744 
0.03  719 
0.03693 

9.86763 
9.86752 
9.86  740 
9.86  728 
9.86717 

XX 
12 
12 
XX 
12 

30 

3 

27 
26 

:i 
:I 

.9 

I 
( 
I 

i: 

ii: 

>.i 

[.2 
2.6 

9 

9 

39 

9-83037 
9.83051 
9.8300? 
9.83078 
9.83092 

»4 
14 
13 
14 
14 

9.96332 
9  96357 
9-96383 
9  .  96  408 

9-96433 

25 
26 
25 
25 
26 

0.03  668 
0.03  643 
0.03  617 
0.03592 
c  03  567 

9.86705 
9  .  86  694 
9.86682 
9.86670 
9  86  659 

XX 
12 
13 
II 

25 
24 

23 

22 
21 

.1 

.2 

>3 

1-3 

2.6 

40 

41 
42 

43 

44 

9.83  106 

9.83  I2O 

9  83  133 
9  83  147 
9.83  161 

14 

13 
14 
»4 
*3 

9-96459 
9.96484 
9-96510 

9.96535 
9.96560 

25 
26 
25 
25 
26 

0.03541 
0.03  516 
0.03490 
0.03  465 
0.03440 

9  86647 
9  86635 
9  86  624 
9.86612 
9.86600 

12 
XI 
12 
12 

20 

19 

ii 

•  3 
•4 

:! 

•  7 

3  9 

1 

M 

4S 
<6 

47 
48 

49 

9-83  174 
9.83  188 
9  .  83  202 

9  83215 
9.83229 

»4 
14 
13 
M 

9.96586 
9.96  61  1 
9  .  96  636 
9  .  96  662 
9.96687 

25 
25 
26 
25 

0.03  414 
0.03389 
0.03  364 
0.03338 
0.03313 

9  86  589 
9  86577 
9  .86  565 
9-86554 
9.86542 

12 
12 
II 
12 

15 
14 
13 

12 
II 

.8 
•9 

3 

I 

i 

2 

3-4 
1.7 

XI 

60 

5i 

52 
53 
54 

9.83  242 
9  83256 
9-83270 
9.83283 
9.83297 

14 
14 
»3 
»4 

9.96  712 
9.96738 
9-96763 
9.96  788 
9.96  814 

25 
26 
25 
25 
26 

0.03  288 
0.03  262 
0.03  237 

0.03  212 

0.03  186 

9.86530 
9.86  518 
9.86  507 
9.86495 
9.86483 

12 
II 
12 
12 

10 

I 

.1         1 

.2       2 
-3       I 

\       \ 

.2 

•4 
.6 

^s 

>.o 

I.I 

2.2 

3-3 
4-4 
5-5 

fi 
•? 

59 

9.83310 
9  83324 
983338 
9-8335I 
983365 

14 
14 
13 
14 

9.96839 
9.96864 
9.96890 
9.96915 
9  96  940 

25 

25 
26 
25 

25 

0.03  161 
o  03  136 
0.03  no 
0.03  085 
o  .  03  060 

9.86472 
9  .  86  460 
9.86448 
9.86436 
9.86425 

12 
12 
12 
II 

5 
4 
3 

2 

*       t 

.8     c, 
•9    ic 

.2 

-4 
.6 
>.S 

o.o 

Ii 

9-9 

60 

9.83378 

9.96  966 

0.03034 

9.86413 

0 

L.  Cos. 

d. 

L.  Cotg. 

c.d. 

L.  Tang. 

L.  Sin. 

d. 

/ 

Pro] 

>• 

Pts. 

47° 

LOGARITHMS  OF  SINE,  COSINE,  TANGENT  AND  COTANGENT,  ETC.     73 


1                                       43° 

, 

L.  Sin. 

d. 

L.  Tang. 

c.d. 

L.  Cotg. 

L.  Cos. 

d. 

Prop.  Pts. 

0 

i 

2 

3 
4 

9  83378 
9-83392 
9-83405 
9.83419 

9-83432 

14 
13 

13 
14 

13 
14 
13 

13 
13 

14 
13 
13 
14 

13 
13 
14 

»3 
13 

13 
13 
»3 

13 

«4 
»3 
U 

14 
13 
J3 
13 
13 

13 
13 
'3 

13 
14 
13 
13 
'3 
13 
13 

9.96966 
9.96991 
9.97016 
9.97042 
9.97067 

25 
25 
26 
25 
25 

0.03034 
0.03  009 

O.02  984 
O.O2  958 

0.02  933 

9-86413 
9.86  401 
9-86389 
9-86377 
9.86366 

12 
12 
12 
II 
12 
S3 
S3 
13 
12 
II 
S3 
12 
13 
12 
13 
12 
S3 
IS 
S3 

GO 

ii 

i 

55 
54 
53 
52 

.1 

.2 
•3 

4 

.1 

:i 

-9 
.1 

.2 
•3 
•4 

:l 

•9 
.1 

.2 

•3 
•4 

:I 

•9 
.1 

.2 

•3 

•4 

i 

•9 
.1 

.2       1 

-3    : 

.4    < 

;!  i 

.9   ic 

aff 
2.6 

w 

10.4 

111 

18.2 

20.8 

23  4 

25 
5-0 
7.5 

10.  0 

12.5 

17^5 
20.  o 
22.5 

14 

1:2 
U 

9-8 

II.  2 
12.6 

13 

39 

7$ 

9-1 
10.4 
11.7 

13             II 

[.2       I.I 
Z.4      2.2 

J-6     3-3 
1-8     4-4 

>-°     5-5 
7.2     6.6 

!:*  Ii 

).8     9.9 

I 

8 
9 

9.83446 
9  83459 
9.83473 
9.83486 
9.83500 

9.97092 
9-97  "8 
9-97  143 
9.97  168 

9-97  193 

26 
25 
25 
25 
26 

25 

25 
26 
25 
25 
26 
25 
25 
26 
25 

25 
26 
25 
25 
25 
26 
25 
25 
26 
25 

25 
26 
25 
25 
25 

26 
25 
25 
26 

25 

25 
26 
25 
25 
25 
26 
25 
25 
26 
25 
25 
25 
26 

25 
25 
26 
25 

«5 

25 
26 

o.o2"9o8 

0.02  882 
0.02  857 
O.02  832 
O.O2  807 

9-86354 
9-86342 
9.86330 
9.86318 
9.86306 

10 

ii 

12 
13 
H 

9.83513 

9.83540 
9.83554 
9-83567 

9.97219 

9-97244 
9-97269 

9.97320 

0.02  781 
O.02  756 
0.02  731 
0.02  705 
0.02  680 

9.86295 
9.86283 
9.86271 
9-86259 
9.86247 

50 

42 
48 

ii 

1L. 

21 
22 
23 

24 

9.83581 

9-83594 
9.83608 
9.83  621 
9-83634 

9-97345 
9-97371 
9.97396 
9-97421 
9-97447 

0.02  655 
O.02  629 
O.O2  604 

0.02  579 
0.02  553 

9.86235 
9.86223 
9.86211 
9.86  200 
9.86  188 

45 
44 
43 
42 

9-83648 
9.83661 
9.83674 
9.83688 
9.83  701 

9-97472 
9-97497 

9-97548 
9-97573 

O.O2  528 
0.02  503 
0.02477 
0.02452 
O.O2  427 

9.86176 
9.86  164 
9.86  152 
9.86  146 
9.86  128 

S3 
S3 
S3 
S3 
S3 
13 
13 
S3 
S3 
S3 
S3 
S3 
S3 
12 
S3 
S3 
13 
S3 
12 
S3 
S3 
S3 
12 
12 
12 
13 

13 
12 
S3 
S3 
S3 
S3 
S3 

13 
S3 

S3 
S3 
S3 
S3 

40 

39 
38 

37 

Jl 
35 
34 
33 
32 

? 

29 

9.83715 
9-83728 
9.83741 
9.83755 
9-83  768 

9-97  598 
9-97624 
9.97649 
9.97674 
9.97  700 

O.O2  4O2 
0.02376 
0.02351 
0.02  326 
0.02  300 

9.86  116 
9.86  104 
9.86092 
9.86080 
9.86068 

3« 

32 
i  33 
34 

9.83781 
9  83  795 
9.83808 
9.83821 
9-83834 

9-97  725 
9=97750 

9-97  776 
9.97801 
9.97826 

O.O2  275 
0.02  2JJO 
0.02  224 
O.O2  199 
0.02  174 

9.86056 
9.86044 
9.86032 
9.86020 
9.86008 

30 

3 

11 

35 
36 

39 

9.83848 
983861 
9.83874 
9-83887 
9.83901 

9-97851 
9.97877 
9.97902 
9.97927 
9-97953 

O.O2  149 
0.02  123 
0.02098 
0.02073 
0.02  047 

9.85996 
9.85984 
9.85972 
9.85960 
9.85948 

25 
24 
23 

22 
21 

w 

19 

i! 

40 

42 

43 
44 

45 
46 

i  48 

1  49 

9.83914 
9.83927 
9.83940 

9  83954 
9.83967 

9-97978 
9.98003 
9.98029 
9.98054 
9-98079 

O.O2  O22 

o.oi  997 
o.oi  971 
o.oi  946 
o.oi  921 

9-85936 
9.85924 
9.85912 
9.85  900 
9.85888 

9.83980 

9.f3993 
9  .  84  006 
9  .  84  020 
9  84  033 

9.98  104 
9-98  130 
9-98155 
9.98  180 
9.98  206 

o.oi  896 
o.oi  870 
o.oi  845 
o.oi  820 
o.oi  794 

9.85876 
9.85  864 
9-85851 
9-85839 
9.85827 

15 
13 

12 
II 

50 

i  5' 

1  53 
1  54 

9  84  046 
9  84059 
9  84072 
9-8408$ 
9  .  84  098 

9-98231 
9-98256 
9.98281 
9.98307 
9-98332 

o.oi  769 
o.oi  744 
o.oi  719 
o.oi  693 
o.oi  668 

9.85815 
9-85803 
9.85  791 
9-85  779 
9-85  766 

10 

S 

5 

4 
3 

i 

55 

56 

58 
59 

9.84  112 
9.84  125 
9.84138 
9.84151 
9.84  164 

9  '.98  p| 
9.98408 

9.98458 

o.oi  643 
o.oi  617 
o.oi  592 
o.oi  567 
o.oi  542 

9-85  754 
9-85742 
9-85  730 
9.85  718 
9-85  7o6 

9.84  177 

9.98484 

o.oi  516 

9-85693 

0 

L.  Cos. 

d. 

L.  Cotsr. 

c.d. 

L.  Tang. 

L.  Sin. 

d. 

f 

Prop.  Pts. 

46° 

TABLE  IV. 




44° 

t 

L.  Sin. 

(!. 

L.  Tang. 

e.d. 

L.  Cot£. 

L.  Cos. 

d. 

Proi 

).  Pts. 

0 
I 

2 

3 
4 

9.84  177 
9.84  190 
9-84203 

9.84  210 
9.84229 

»3 
13 

*3 
»3 

9.98484 
9.98509 

9-98534 
9.98  560 

9-98585 

25 
25 
26 
25 

o.oi  516 
o.oi  491 

O.OI  466 

o.oi  440 
o.oi  415 

9-85693 
9.85  68  1 
9-85669 
9-85657 
9-85645 

12 
12 

xa 

12 

(>0 

9 

11 

,i 

26 

2.6 
1    2 

I 
I 

9 

9  .  84  242 
9  84255 
9  .  84  269 
9  84282 
9.84295 

*3 
»4 
>3 
*3 

9.98  610 
9-98635 
9.98661 
9.98686 
9.98711 

25 
26 
25 
25 
26 

o.oi  390 
o.oi  365 
o.oi  339 
o.oi  314 
o.oi  289 

9.85632 
9.85  620 
9.85608 
9.85596 
9.85  583 

12 
12 
12 
»3 

55 
54 
53 
52 
5i 

•3 
•4 

:I 

.7 

f:« 

10.4 
13.0 
15.6 
18.2 

10 

ii 

12 
*3 

H 

9.84308 
9.84321 
9-  ^4334 
9.84347 
9.84360 

»3 
»3 
13 
»3 

9.98737 
9.98  762 

9-98  787 
9.98812 
9.96838 

25 
25 
«s 
26 

o.oi  263 
o.oi  238 
o.ci  213 
o.oi  188 
o.oi  162 

9.85571 
9.85559 
9-85  547 
9.85534 
9-85  522 

12 
12 

13 
12 

50 

49 
48 

47 
46 

.8 
•9 

20.8 

23.4 

25 

3- 

!1 

19 

9.84373 
9.84385 
9.84398 
9.84411 
9.84424 

12 

»3 

»3 
13 

9.98863 
9.98808 
9.98913 

9  9*939 

9.98964 

25 
25 
26 
25 

o.oi  137 

O.OI   112 

o.oi  087 
o.oi  obi 
o.oi  036 

9.85  510 

9.85497 
9.85485 

9-85473 
9.85460 

«3 
12 
12 
»3 

45 
44 
43 
42 
41 

.1 

.2 

•3 
•4 

I 

2.5 

5-0 
7-5 

10.  0 

12.5 

20 

21 

22 
23 
24 

9-84437 
9.84450 
9.84463 
9.84476 
9.84489 

»3 
»3 
J3 
13 

9.98989 
9.99015 
9.99040 
9.99065 
9.99090 

26 
25 
25 
25 
26 

O.OI  Oil 

0.00985 
o.oo  960 

0.00935 

0.00910 

9-85448 
9-85436 
9-85423 
9.85411 

9.85399 

12 

'3 

12 
12 

40 

I 

.0 

:1 

•9 

IS  ° 

17-5 

20.0 
22.5 

S 

27 
28 
29 

9-8450^ 
9.84515 
9-84528 
9.84540 

9.84553 

13 
*3 

12 
13 

9.99  116 

9-99  HI 
9.99  166 

9-99  191 
9.99217 

25 
25 

85 

26 

0.00884 
0.00859 
o.oo  834 
0.00809 
o.oo  783 

9-85386 
9-85374 
9-85361 
9  85349 
9-85337 

19 
<3 
12 

12 

35 
34 
33 
32 
3i 

.1 

.2 
•3 
•4 

M 

5:1 

4.2 
5-6 

80 

3» 

32 
33 
34 

9-84566 

9  84579 
9  84592 
9.84605 
9.84618 

«3 
«3 
«3 
*3 

9-99242 
9.99267 

9-99293 
9.99318 

9-99343 

25 
26 

25 
25 

o.oo  758 
o.oo  733 
o.oo  707 
0.00682 
0.00657 

9-85  324 
9-85312 

9-85  299 
9-85287 
9.85274 

12 

«3 

12 

«3 

12 

80 

1 

:i 

i 

•9 

7-0 
8-4 
9.8 

II.  2 
12.6 

9 
9 

39 

9  84  630 
9.84  643 
9  .  64  656 
9.84669 
9.84082 

*3 
«3 
»3 
»3 

9.99368 
9-99394 
9.99419 
9.99444 
9.99469 

26 
25 
25 

25 
26 

0.00632 
0.00606 
o.oo  581 
0.00556 
o.oo  531 

9  .  85  262 
9-85250 
9-85237 
9-85  225 

9.85  212 

12 

«3 

12 

>3 

12 

25 
24 

23 

22 
21 

.1 

.2 

«3 

11 

40 

41 
42 

43 

44 

9-84694 
9-84707 
9.84720 

9  84733 
9-»4  745 

«3 
»3 
13 

'    12 

9-99495 
9-99  520 
9-99545 
9-99570 
9-99  59& 

25 
25 
25 
26 

o.oo  505 
o.oo  480 
o.oo  455 
0.00430 
0.00404 

9.85  200 

9.85  187 

9-85  175 
9.85  162 

9-85  15° 

»3 

12 

«3 

12 

i>0 

;i 

3 

•3 

!i 

•7 

39 

l:i 

7-8 
9-i 

3 
J2 

49 

9-84758 
9.84  771 
9.84  784 
9.84  796 
9  .  84  809 

X3 
13 
13 

12 

«3 

9.99621 
9.99646 
9.99672 
9.99697 
9.99722 

8S 

25 

26 

25 

25 

0.00379 
0.00354 
0.00328 
0.00303 
0.00278 

9-85  137 
9-85  125 
9.85  112 

9.85  ioo 

9.85087 

12 

*3 

12 
13 

15 
14 
13 

12 
II 

.8 
•9 

10.4 
11.7 

xa 

50 

5i 
52 
53 

54 

9.84822 
9.8483$ 
9-84847 
9.84860 

9-84873 

X3 
«3 

12 

«3 
«3 

9-99747 
9-99  773 
9.99798 
9-9982;? 
9.99848 

25 

26 

25 

25 
25 

0.00253 
0.00227 

O.OO  2O2 

o.oo  177 
o.oo  152 

9.85074 
9.85062 

9.85049 
9.85037 
9.85024 

18 

«3 

12 
>3 

10 

1 
I 

.1 

.2 

•3 
•4 

•1 

1.2 

'I 
tt 

P 
12 

59 

9-84885 
9.8489$ 
9.849" 
9.84923 
9.84936 

»3 
»3 

19 

«3 

9.99874 
9-99899 
9-99924 
9.99949 
9-99975 

25 
25 
25 

96 

o.oo  126 

0.00  101 

0.00076 
o.oo  051 
0.00025 

9.85012 

9.84999 
9.84986 

9.84974 
9.84961 

«3 
«3 

19 

»3 

ft 

5 

4 
3 

i 

.0 

i 

•9 

*1 

,2:1 

00 

9  84949 

*3 

o.ooooo 

25 

o.oo  ooo 

9.84949 

0 

L.  Cos. 

d. 

L.  Cotg. 

c.d. 

L.  Tang:. 

L.  Sin. 

d. 

i 

Pro] 

[>.  PtS. 

45° 

TABLE  V— NATURAL  SINES  AND  COSINES.  75 


TABLE  V. 


NATURAL 


SINES  AND  COSINES 


)                                             1  Alil^r*  V  • 

0° 

1° 

2° 

3° 

40 
40 

60 
59 

58 

H 

55 
54 

53 
52 
5i 
50 

2 

t 

tf.  sine 

N.  cos. 

N.  sine 

ST.  cos. 

N.  sine 

N.  cos. 

N.  sine 

^.  cos. 

N.  sine|N.  cos. 

O 

I 

2 

3 
4 

I 
'I 

9 

1C 

ii 

12 

~^3~ 

14 

II 
1 

19 

20 
21 
22 

23 
24 

.00000 

.00029 
.00058 

.00087 

.00116 

.00145 
.00175 

.00000 
.00000 

.00000 
.00000 

.00000 
.00000 
.00000 

•01745 
.01774 
.01803 
.01832 
.01862 
.01891 
.01920 

.99985 
.99984 
.99984 
.99983 
.99983 
.99982 
.99982 

.03490 
•03519 
•03548 
-03577 
.03606 

•03635 
.03664 

•99939 
.99938 

•99937 
•99936 
•99935 
•99934 
•99933 

•35234 
.05263 
.05292 
•05321 
•05350 
•05379 
.05408 

.99863 

99861 
.99860 

.99858 
•99857 
•99855 
•99854 

.06976 
.07005 
.07034 
.07063 
.07092 
.07121 
.07150 

•99756 
•99754 
•99752 
•99750 
•99748 
•99746 
•99744 

.00204 
.00233 
.00262 
.00291 

.00320 
.00349 

.00000 
.00000 
.00000 
.00000 

•99999 
•99999 

.01949 
.01978 
.02007 
.02036 
.02065 
.02094 

.99981 
.99980 
.99980 
•99979 
•99979 
•99978 

.03693 
.03723 
•03752 
.03781 
.03810 
•03839 

•99932 
•99931 
.99930 
.99929 

•99927 
.99926 

•°5437 
.05466 

•05495 
•05524 
•05553 
.05582 

.99852 
.99851 

•99849 
.99847 
.99846 
.99844 

.07179 
.07208 
.07237 
.07266 
.07295 
•07324 

.99742 
.99740 
•99738 
•99736 
•99734 
•99731 

.00378 
.00407 
.00436 
.00465 
.00495 
.00524 

•99999 
•99999 
•99999 
•99999 
•99999 
•99999 

.02123 
.02152 
.02181 

.02211 

.O224O 
.02269 

•99977 
•99977 
.99976 

•99976 
•99975 
•99974 

.03868 
.03897 
.03926 

•03955 
.03984 
.04013 

•99925 
.99924 
.99923 
.99922 
.99921 
.99919 

.05611 
.05640 
.05669 
.05698 
•05727 
•05756 

.99842 
.99841 

•99839 
.99838 

•99836 
•99834 

•07353 
•07382 
.07411 
.07440 
.07469 
.07498 

.99729 
.99727 
•99725 
•99723 
•99721 
.99719 

9 

45 
44 
43 
42 

•005*3 
.00582 
.00611 
.00640 
.00669 

.00698 

•99998 
.99998 
.99998 
•99998 
.99998 
.99998 

.02298 
.02327 
•02356 
•02385 
.02414 
.02443 

•99974 
•99973 
•99972 
.99972 
.99971 
.99970 

.04042 
.04071 
.04100 
.04129 
.04159 
.04188 

.99918 
.99917 
.99916 

•99915 
•99913 
.99912 

•05785 
•05814 
.05844 

•05873 
.05902 

•05931 

•99833 
.99831 
.99829 
.99827 
.99826 
.99824 

.07527 
•07556 
•07585 
.07614 

•07643 
.07672 

.99716 
.99714 
.99712 
.99710 
.99708 
•99705 

41 
40 

39 
38 

37 

36 

2 

3 

29 
jp_ 

3i 
32 
33 
34 
35 
_36_ 

11 

39 
40 

4i 
42 

43 
44 
45 
46 

47 
48 

.00727 
.00756 
.00785 

.00814 

.00844 
.00873 

•99997 
•99997 
•99997 
•99997 
.99996 
.99996 

.02472 
.02501 
.02530 
.02560 
.02589 
.02618 

•99969 

.99967 
.99966 
.99966 

.04217 
.04246 
.04275 
.04304 

•04333 
.04362 

.99911 
.99910 
.99909 
.99907 
.99906 
.99905 

.05960 

•05989 
.06018 
.06047 
.06076 
.06105 

.99822 
.99821 
.99819 

•99817 
.99815 
.99813 

.07701 
.07730 

•07759 
.07788 
.07817 
.07846 

•99703 
•99701 
.99699 
.99696 

•99694 
.99692 

35 

34 
33 
32 
3i 
30 

.00902 

.00931 

.00960 

.00989 

.01018 

.01047 

.99996 
.99996 
•99995 
•99995 
•99995 
•99995 

.02647 
.02676 
.02705 
.02734 
.02763 
.02792 

•99965 
.99964 

•99963 
.99963 
.99962 
.99961 

.04391 
.04420 

•04449 
.04478 

•04507 
•04536 

•99904 
.99902 
.99901 

•99897 

.06134 
.06163 
.06192 
.06221 
.06250 
.06279 

1.99812 
.99810 
.99808 
.99806 
.99804 
•99803 

•07875 
.07904 

•07933 
.07962 
.07991 
.08020 

.99689 
.99687 
.99685 
•99683 
.99680 
.99678 

% 

11 

25 

24 

.01076 
.01105 
.01134 

.01164 
.01193 

.01222 

.99994 
•99994 
•99994 
•99993 
•99993 
•99993 

.02821 
.02850 
.02879 
.02902 
.02938 
.02967 

.99960 
•99959 
•99959 
.99958 

•99957 
•99956 

•04565 
.04594 
.04623 

•04653 
.04682 
.04711 

.99896 
.99894 

•99893 
.99892 

.06308 
.06337 
.06366 

•06395 
.06424 
•06453 

.99801 

•99799 
•99797 
•99795 
•99793 
.99792 

.08049 
.08078 
.08107 
.08136 
.08165 
.0819^ 

.99676 

•99673 
.99671 
.99668 
.99666 
.99664 

23 

22 
21 

2O 

*9 

.01251 
.01280 
.01309 
•01338 
.01367 
.01396 

•99992 
•99992 
.99991 
.99991 
.99991 
.99990 

.02996 
.03025 

03054 
.03083 
.O3II2 
.03141 

•99955 
•99954 
•99953 
.99952 

•99952 
•99951 

.04740 
.04769 
.04798 
.04827 
.04856 
.04885 

.99888 
.99886 
.99885 
.99883 
.99882 
.99881 

.06482 
.06511 
.06540 
.06569 
.06598 
.06627 

.99790 
.99788 
•99786 
.99784 
•99782 
.99780 

.08223 
.08252 
.08281 
.08310 

.0836* 

.99661 
.99659 

•99657 
.99654 
.99652 
•99649 

II 

15 
H 
13 

12 
II 

10 

I 

49 
50 
5i 

!  52 

,  53 
5* 

.01425 
.01454 
.01483 
.01513 
.01542 
.01571 

•99990 
•99989 
•99989 
.99989 
.99988 
.99988 

.03170 
.03199 
.03228 

•03257 
.03286 
.03316 

.99950 
•99949 
•99948 
•99947 
•99946 
•99945 

.04914 
.04943 
.04972 
.05001 
•05030 
•05059 

.99879 
.99878 
.99876 
•99875 
•99873 
•99872 

.06656 
.06685 
.0671^ 
•06743 
.06773 
.06802 

.99778 
•99776 
•99774 
•99772 
.99770 
.99768 

.08397 
.08426 
.08455 
.08484 

S&3 

•99647 
.99644 
•99642 
•99639 
.99637 
•99635 

1  55 
56 
57 
5« 

8 

.OI6OO 
.01629 
.01658 
.01687 
.01716 
01745 

N.  cos. 

.99987 
.99987 
.99986 
.99986 
.99985 
.99985 

N.  sine 

•03345 
•03374 
•03403 
•03432 
.03461 
.03490 

N.  cos. 

•99944 
•99943 
.99942 
.99941 
.99940 
•99939 

N.  sine 

.05088 
.05117 
.05146 

•05175 
.05205 

•05234 
N.  cos. 

.99870 
.99869 
.99867 
.99866 
.99864 
•99863 

N.  sine 

.06831 
06860 
.00889 
.06918 
.06947 
.06976 

N.  cos. 

.99766 
.99764 
.99762 
.99760 
•99758 
•99756 

N.  sine 

S£ 

.08629 
.08658 
.08687 
.08716 

N.  cos. 

1.99632 
.99630 
.90627 
•99625 
.99622 
.99619 

N.  sine 

5 
4 
3 

2 
I 
0 

f 

89°' 

88° 

87° 

86° 

85° 

NATURAL  SINES  AND  COSINES. 


77 


5° 

6° 

7° 

8° 

9° 

/ 

o 

I 

2 

3 
4 

1 

N.  sine 

N.  cos. 

N.  sine 

N.  cos. 

N.  sine 

N.  cos. 

N.  sine 

N.  cos. 

N.  sine 

N.  cos. 

.08716 

.08745 
.08774 
.08803 
.08831 
.08860 
.08889 

.99619 
.99617 
.99614 
.99612 
.99609 
.99607 
.99604 

•  10453 
.  10482 
.10511 

•  10540 
.10569 

.10597 
.  10626 

.99452 
•99449 
•99446 

•99443 
.99440 

•99437 
•99434 

.12187 
.12216 
.12245 
.12274 
.12302 

•12331 
.12360 

•99255 
.99251 
.99248 

•99244 
.99240 
.99237 
•99233 

•I39I7 
.13946 

•13975 
.14004 

•14033 
.14061 
.  14090 

.99027 
.99023 
.99019 
.99015 
.99011 
.99006 
.99002 

•15643 
.15672 
.15701 
•15730 
•15758 
•15787 
.15816 

.98769 
.98764 
.98760 
•98/55 
•9875' 
.98746 
.98741 

60 

9 

9 

55  ' 

54  , 

I 

9 

10 

ii 

12 

.08918 
.08947 
.08976 
.09005 
.09034 
.09063 

.99602 

•99599 
.99596 

•99594 
•99591 
.99588 

!  10684 
.10713 
.10742 
.10771 
.10800 

•99431 
.99428 
.99424 
.99421 
.99418 
•99415 

.12389 
.12418 
.12447 
.12476 
.12504 
•12533 

•99230 
.99226 
.99222 
.99219 

•99215 
.99211 

.14119 
.14148 
.14177 
.14205 
.14234 
.14263 

.98998 

•98994 
.98990 
.98986 
.98982 
.98978 

•15845 
•15873 
.15902 

•I593I 
•15959 
.15988 

•98737 
•98732 
.98728 
.98723 
.98718 
.98714 

53  i 
52 
Si 
50 
49 
48 

13 
14 
15 

11 

.09092 
.09121 
.09150 
.09179 
.09208 
.09237 

.99586 

•99583 
.99580 

•99578 
•99575 
•99572 

.  10829 
.  10858 
.  10887 
.10916 
.10945 
.10973 

.99412 
.99409 
.99406 
.99402 
•99399 
•99396 

.12562 
.12591 
.12620 
.12649 
.12678 
.12706 

.99208 
.99204 
.99200 
.99197 

•99193 
.99189 

.14292 
.14320 
•14349 
•14378 
.I44CJ 
.14436 

•98973 
.98969 
.98965 
.98961 
•98957 
•98953 

.16017 
.16046 
.16074 
.16103 
.16132 
.16160 

.98709 
.98704 
.98700 
.98695 
.98690 
.98686 

47 
46 

45 
44 
43 
42 

19 
2O 
21 
22 

23 
24 

.09266 
.09295 
.09324 

•09353 
.09382 
.09411 

•9957° 
•99567 
.99564 
.99562 
•99559 
•99556 

.IIOO2 
.11031 
.11060 
.11089 
.IIIlS 
.11147 

•99393 
.99390 
.99386 

•99383 
•99380 
•99377 

•12735 
.12764 
.12793 
.12822 
.12851 
.12880 

.99186 
.99182 
.99178 

•99175 
.99171 
.99167 

.14464 

.14493 
.14522 

•I455I 
.14580 
.14608 

.98948 
.98944 
.98940 
.98936 
.98931 
.98927 

.16189 
.16218 
.16246 
.16275 
.16304 
•16333 

.98681 
.98676 
.98671 
.98667 
.98662 
.98657 

41 
40 

39 
38 
37 
36 

2 

3 

29 
30 

.09440 
.09469 
.09498 
.09527 
•09556 
•09585 

•99553 
•99551 
.99548 

•99545 
•99542 
.99540 

.  1  1  1  76 
.11205 
.11234 
.11263 
.11291 
.II32O 

•99374 
•99370 
•99367 
•99364 
•9936o 
•99357 

.12908 

•12937 
.12966 
.12995 
.13024 
•13053 

.99163 
.99160 
.99156 
.99152 
.99148 
.99144 

•14637 
.14666 
.14695 
•H723 
•14752 
.14781 

.98923 
.98919 
.98914 
.98910 
.98906 
.98902 

.16361 
.16390 
.16419 
.16447 
.16476 
•16505 

.98652 
.98648 
•98643 
.9863$ 

•98633 
.98629 

35 
34 
33 
32 
31 
30 

31 
32 
33 
34 

I 

.09614 
.09642 
.09671 
.09700 
.09729 
.09758 

•99537 
•99534 

•99531 
.99528 

•99526 
•99523 

•II349 
.11378 
.11407 
.11436 
.11465 
.11494 

•99354 
•99351 
•99347 
•99344 
•99341 
•99337 

.13081 
.13110 

•I3J39 
.13168 

•I3I97 
.13226 

.99141 
•99137 
•99133 
.99129 
.99125 
.99122 

.14810 
•14838 
.14867 
.14896 
.14925 
•14954 

.98897 

!  98884 
.98880 
.98876 

•16533 
.16562 

•'SB! 

.16620 

.16648 
.16677 

.98624 
.98619 
.98614 
.98609 
.98604 
.98600 

% 

11 

25 
24 

9 

39 
40 

4i 
42 

.09787 
.09816 
.09845 
.09874 
.09903 
.09932 

.99520 
•99517 
•99514 
•995" 
.99508 
.99506 

•H523 

•"552 

.11580 
.11609 
.11638 
.11667 

•99334 
•99331 
•99327 
•99324 
.99320 

•99317 

•13254 
•13283 
•I33I2 
•I334I 
•13370 
•13399 

.99118 
.99114 
.99110 
.99106 
.99192 
.99098 

.14982 
.15011 
.15040 
.15069 

.15097 
.15126 

.98871 
.98867 
.98863 
.98858 
.98854 
.98849 

.16706 

•16734 
.16763 
.16792 
.16820 
.16849 

•98595 

•98585 
.98580 

•98575 
.98570 

23 

22 
21 
20 

43 

44 

9 
3 

.09961 
.09990 
.10019 
.10048 
.10077 
.10106 

•99503 
.99500 

•99497 
•99494 
.99491 
.99488 

.11696 
.11725 

.11754 
.11783 
.11812 
.11840 

•993M 
.99310 

•99307 
•99303 
.99300 

•99297 

•13427 
13456 
•13485 
•I35I4 
•13543 
•13572 

•99094 

.99083 
•99079 
•99075 

•I5I55 
.15184 
.15212 
.15241 
.15270 
.15299 

•98845 
.98841 
.98836 
.98832 
.98827 
.98823 

.16878 
.16906 

•'6935 
.1696^ 
.16992 
.17021 

•98565 
.98561 
•98556 
•98551 
.98546 
.98541 

\l 

15 
H 
13 
12 

49 
50 
51 

52 
53 
54 

•IOI35 
.10164 
.10192 
.10221 
.10250 
.10279 

.99485 
.99482 
•99479 
•99476 

•99473 
.99470 

.11869 
.11898 
.11927 
.11956 
.11985 
.12014 

•99293 
.99290 
.99286 
•99283 
•99279 
-99276 

.13600 
.13629 
•13658 
•13687 
.13716 

•13744 

.99071 

.99067 
.99063 
.99059 

•99055 
.99051 

•15327 
•15356 
•15385 
•I54H 
.15442 

•I547I 

.98818 
.98814 
.98809 
.98805 
.98800 
.98796 

.17050 
.17078 
.17107 
.17136 
.17164 
•17193 

•98536 
•98531 
•98526 
.98521 
•98516 
.98511 

11 

10 

I 

55 
56 

9 

S 

.  10308 

•10337 
.10366 
•10395 
.10424 
•10453 

.99467 
.99464 
.99461 
.99458 
•99455 
•99452 

.12013 
.12071 

.12100 
.12129 
.12158 
.12187 

.99272 
.99269 
•99265 
.99262 
.99258 
99255 

•13773 
.13802 

:& 

•13889 
13917 

.99047 

•99043 
.99039 

•99035 
.99031 
.99027 

.15500 
•15529 
•15557 
.15586 
.15615 
•15643 

.98791 
.98787 
.98782 
•98778 

•98773 
.98769 

.  i  7222 
•17250 
.17279 
.17308 
•17336 
•17365 

.98506 
.98501 
.98496 
.98491 
.98486 
.98481 

5 
4 
3 

2 
I 
O 

( 

N.  cos. 

N.  sine 

N.  cos. 

N.  sine 

N.  cos. 

N.  sine 

N.  cos. 

N.  sine 

N.  cos. 

N.  sine 

9 

84° 

83° 

82° 

81° 

80° 

78                                                            TABLE  V. 

10° 

11°    % 

12° 

13° 

14° 

/ 

N.  sine 

N.  cos. 

N.  sine 

N.  cos. 

N.  sine 

N.  cos. 

N.  sine 

N.  cos. 

N.  sine 

N.  cos. 

o 

i 

2 

3 

1 

•17365 
•17393 

.17422 

•I745I 
•17479 
.17508 

•17537 

.98481 
.98476 
.98471 
.98466 
1.98461 
.9f455 
•9845° 

.19081 
.19109 
.19138 
.19167 

.19195 
.19224 
.19252 

•98163 

•98i57 
.98152 
.98146 
.98140 

•98135 
.98129 

.20791 
.20820 
.20848 
.20877 
.20905 
.20933 
.20962 

•978i5 
.97809 
.97803 

•97797 
.97791 

•97784 
•97778 

.22495 
.22523 
•22552 
.22580 

.22637 
.22665 

•97437 
•9743° 
•97424 
•97417 
.97411 

•97404 
•97398 

.24192 
.24220 
.24249 
.24277 
•24305 
•24333 
.24362 

.97030 
.97023 

•97015 
.97008 
.97001 

60 
59 
58 

!? 

55 
54 

I 

9 

10 

ii 

12 

•17565 
•17594 
.17623 

:$£ 

.17708 

•98445 
.98440 

•98435 
.98430 
.98425 
.98420 

.19281 
.19309 
•I9338 
.19366 

•19395 
.19423 

.98124 
.98118 
.98112 
.98107 
.98101 
.98096 

.20990 
.21019 
.21047 
.21076 
.21104 
.21132 

.97772 
.97766 
.97760 
•97754 
•97748 
.97742 

.22693 
.22722 
.22750 
.22778 
.22807 
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•97391 
•97384 
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.24390 
.24418 
.24446 
•24474 
•24503 
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53 
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50 

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:!$67 

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.98414 
.98409 
.98404 
.98399 
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.19452 
.19481 
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.I953f 
.19566 
•19595 

.98079 

•98073 
.98067 
.98061 

.21161 
.21189 
.21218 
.21246 
.21275 
.21303 

•97735 
•97729 
•97723 
.97717 
.97711 
•97705 

.22863 
.22892 
.22920 
.22948 
.22977 
.23005 

•97351 
•97345 
•9733s 
•97331 
•97325 
•97318 

•24559 
.24587 
.24615 
.24644 
.24672 
.24700 

.96937 

.96930 
.96923 
.96916 
.96909 
.96902 

9 

45 
44 
43 
42 

19 
20 

21 
22 

23 

24 

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2 

29 

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•17937 
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.98362 
•98357 

.19623 
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.19737 
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•98033 
.98027 

•21331 
.21360 
.21388 
.21417 

•21445 
.21474 

.97698 
.97692 

.97680 

•97673 
.97667 

•23033 
.23062 
.23090 
.23118 
.23146 
•23175 

•973H 
•97304 
.97298 

•97291 
.97284 
•97278 

.24728 
•24756 
•24784 
.24813 
.24841 
.24869 

.96894 
.96887 
.96880 

•96873 
.96866 
.96858 

41 
40 

P 

11 

.18081 
.18109 
.18138 
.18166 
•18195 
.  18224 

•98352 
•9f347 
.98341 

•98336 
•98331 
•98325 

.19794 
.19823 
.19851 
.19880 
.19908 
•19937 

.98021 
.98016 
.98010 
.98004 
•97998 
•97992 

.21502 
•21530 

•21559 
.21587 
.21616 
.21644 

.97661 

2$ 
3& 

.97630 

.23203 
.23231 
.23260 
.23288 
.23316 
•23345 

.97271 
.97264 
.97257 
•97251 
•97244 
•97237 

.24897 
•24925 
•24954 
.24982 
.25010 
.25038 

.96851 
.96844 
.96837 
.96829 
.96822 
•96815 

35 
34 
33 
32 
31 
30 

31 
32 

33 
34 

$ 

.18252 
.18281 
.18309 
•18338 
•18367 
.18395 

.98320 

•98315 
.98310 
.98304 
.98299 
.98294 

.19965 
.19994 

.20022 
.20051 
.20079 
.20108 

.97987 
.97981 
•97975 
•97969 
.97963 
•97958 

.21672 
.21701 
.21729 
.21758 
.21786 
.21814 

.97623 
.97617 
.97611 
.97604 
•97598 
•97592 

.23373 
.23401 
.23429 
•23458 
.23486 

•23514 

.97230 
•97223 
.97217 
.97210 
.97203 
.97196 

.25066 
.25094 
.25122 
.25151 
•25179 
.25207 

.96807 
.96800 

.96793 
.96786 
.96778 
.96771 

1 

25 

24 

S 

39 
40 

4i 

42 

.18424 
.18452 
.18481 
.18509 
•18538 
•18567 

.98288 
.98283 
.98277 
.98272 
.98267 
.98261 

.20136 
.20165 
.20193 
-2O222 
.20250 
.20279 

•97952 
.97946 
•97940 
•97934 
.97928 
.97922 

.21843 
.21871 
.21899 
.21928 
.21956 
.21985 

•97585 
•97579 
•97573 
•97566 
•97560 
•97553 

.23542 
.23571 
•23599 
.23627 
.23656 
.23684 

.97189 
.97182 
.97176 
.97169 
.97162 
.97155 

•25235 
.25263 
•25291 
•25320 
•25348 
•25376 

.96764 
.967.56 
.96749 
.96742 
•96734 
•96727 

23 

22 
21 

2O 

43 

44 
i  45 
46 

2 

•18595 
.18624 
.18652 
.18681 
.18710 
.18.738 

.98256 
.98250 

•98245 
.98240 
.98234 
.98229 

.20307 
.20336 
20364 
•20393 
.20421 
.20450 

.97916 
.97910 
•97905 
•97899 
•97893 
•97887 

.22013 
.22041 
.22070 
.22098 
.22126 
•22155 

•97547 
•97541 
•97534 
•97528 
•97521 
•97515 

.23712 

.23740 
.23769 
.23797 
.23825 
•23853 

.97148 
.97141 

•97134 
.97127 
.97120 
•97"3 

•25404 
•25432 
.25460 
.25488 
.25516 
•25545 

.96719 
.96712 
.96705 
.96697 

\l 

15 

H 
13 

12 

49 
50 
5i 
52 
53 
54 

.18767 

•i8795 
.18824 
.18852 
.18881 
.18910 

.98223 
.98218 
.98212 
.98207 
.98201 
.98196 

.20478 
.20507 

•20535 
.20563 
.20592 
.2O62O 

.97881 

•97875 
.97869 
.97863 
•97857 
•97851 

.22183 

.22212 
.22240 
.22268 
.22297 
•22325 

.97508 
.97502 
.97496 
•97489 
•97483 
•97476 

.23882 
.23910 

20 

•23995 
.24023 

.97106 
.97100 

•97093 
.97086 

.97079 
.97072 

•25573 
.25601 
.25629 

•25657 
.25685 

•25713 

•96075 
.96667 
.96660 
.96653 
.96645 
.96638 

II 
10 

I 

I 

55 
56 

H 

8 

.18938 
.18967 
.18995 
.19024 
.19052 
.19081 

N.  cos. 

.98190 
.98185 
.98179 
.98174 
.98168 
•98163 

N.  sine 

.20649 
.20677 
.20706 
•20734 
•20763 
.20791 

N.  cos. 

•97845 
•97839 
•97833 
•97827 
.97821 
•978i5 

N.  sine 

•22353 
.22382 
.22410 
.22438 
.22467 
•22495 

N.  cos. 

•97470 
•97463 
•97457 
.97450 
•97444 
•97437 

N.  sine 

.24051 
.24079 
.24108 
.24136 
.24164 
.24192 

N.  cos. 

.97065 
.97058 

.97051 
.97044 

.97037 
•97030 

N.  sine 

•25741 
.25769 

.25826 

-25854 
.25882 

N.  cos. 

.96630 

•'12 

[96600 
•96593 

M.  sine 

5 
4 
3 

2 
O 
/ 

79° 

7§° 

77° 

76° 

75° 

NATURAL  SINES  AND  COSINES. 


79 


15° 

16° 

17° 

18° 

19° 

t 

N".  sine 

N.  cos. 

N.  sine 

N.  cos. 

N.  sine 

N.  cos. 

N.  sine 

N.  cos. 

N.  sine 

N.  cos. 

o 
I 

2 

3 
4 

.25882 
.25910 

•25938 
.25966 

•25994 
.26022 
.26050 

.96593 
.96585 
.96578 
.96570 
.96562 
•96555 
•96547 

•27564 
.27592 
.27620 
.27648 
.27676 
.27704 
.27731 

.96126 
.96118 
.96110 
.96102 

^96078 

29237 
.29265 
•29293 
•29321 
.29348 
.29376 
.29404 

•95630 
.95622 

•95613 
•95605 
•955?6 
.95588 
•95579 

.30902 
.30929 
•3°957 
•30985 
.31012 
.31040 
.31068 

.95106 

•95097 
.95088 

•95079 
•95070 
•95061 
•95052 

•32557 
•32584 
.32612 
.32639 
.32667 

•32694 
.32722 

•94552 
•94542 
•94533 
•94523 
•94514 
•94504 
•94495 

60 

59 
58 

11 

55 
54 

9 

10 

ii 

12 

.26079 
.26107 
•26135 
.26163 
.26191 
.26219 

.96540 

•96532 
.96524 

•96517 
.96509 
•96502 

•27759 
.27787 
.27815 
.27843 
.27871 
.27899 

.96070 
.96062 
.96054 
.96046 

•96037 
.96029 

.29432 
.29460 
.29487 
•29515 
•29543 
•29571 

•95571 
•95562 
•95554 
•95545 
•95536 
•95528 

•31095 
•3II23 
.31151 
.31178 
.31206 
•3^233 

•95°43 
•95033 
.95024 

•95015 
.95006 

94997 

•32749 

.32777 
.32804 
.32832 

•32859 
•32887 

.94485 
.94476 
.94466 
•94457 
•94447 
.94438 

53 
52 
5i 
50 

4J 

13 
14 

ii 

|| 

.26247 
.26275 
.26303 
-26331 
•26359 
.26387 

.96494 
.96486 
.96479 
.96471 
.96463 
.96456 

.27927 

•27955 
.27983 
.28011 
.28039 
.28067 

.96021 
.96013 
.96005 

•95997 
.95989 
.95981 

•2Q5C9 
.29626 
.29654 
.29682 
.29710 
•29737 

•95519 
•955" 
•95502 

•95493 
•95485 
•95476 

.31261 
.31289 
•31316 
•31344 
•31372 
•31399 

.94988 
•94979 
•94970 
.94961 
•94952 
•94943 

.32914 
•32942 
.32969 
•32997 
•33024 
•33051 

.94428 
.94418 
•94409 
•94399 
.94390 
.94380 

9 

45 
44 
43 
42 

19 

20 
21 
22 

»3 

24 

•26415 
.26443 
.26471 
.26500 
.26528 
.26556 

.96448 
.96440 
•96433 
•96425 
.96417 
.96410 

.28095 
.28123 
.28150 
.28178 
.28206 
.28234 

•95972 
•95964 
•95956 
.95948 
.95940 
•95931 

•29765 
•29793 
.29821 
.29849 
.29876 
.29904 

•95467 
•95459 
•9545° 
•95441 
•95433 
.95424 

•31427 
•3H54 
.31482 
•3i5io 
•31537 
•31565 

•94933 
•94924 
•94915 
.94906 

.94897 
.94888 

•33079 
.33106 

•33!34 
•33161 
•33189 
.33216 

•94370 
.94361 

•94351 
•94342 
•94332 
•94322 

41 
40 

9 

H 

% 

27 

28 
29 
3° 

.26584 
.26012 

'.2.6696 
.26724 

.96402 
.96394 
.96386 

•96379 
.96371 

•96363 

.28262 
.28290 
.28318 
•28346 

•28374 
.28402 

•95923 
•95915 
•95907 
.95898 
.95890 
.95882 

.29932 
.29960 
.29987 
•30015 
•30043 
.30071 

•95415 
•95407 
•95398 
•95589 
•95380 
•95372 

•31593 
.31620 
.31648 
•3l675 
•31703 
•31730 

.94878 
.94869 
.94860 
.94851 
.94842 
.94832 

•33JM4 

•332/1 
•3329? 
•33326 
•33353 
•33381 

•94313 
•94303 
•94293 
.94284 
.94274 
.94264 

35 
34 
33 
32 
3i 
30 

3i 

32 
33 
34 

i 

.26752 
.26780 
.26808 
.26836 
.26864 
.26892 

•96355 
•96347 
.96340 
.96332 
.96324 
.96316 

.20429 

•28457 
.28485 

•2C5I3 
.28541 
.28569 

•95*74 
.95865 

•95857 
.95849 
.95841 
•95832 

.30098 
.30126 

•3OI54 
.30182 
.30209 
.30237 

•95363 
•95354 
•95345 
•95337 
•95328 
•95319 

•31758 
•31786 
•31813 
.31841 
.31868 
.31896 

.94823 
.94814 
.94805 

•94795 
.94786 

•91-777 

•33408 
•33436 
•33463 
•33490 
•33518 
•33545 

94254 
•94245 
•94235 
•94225 

•94215 
.94206 

11 
3 

25 

24 

9 

39 
40 

41 

1  42 

.26920 
.26948 
.26976 
.27004 
.27032 
.27060 

.96308 
.96301 
•96293 
•96285 
.96277 
.96269 

.28597 
.28625 
.28652 
.28680 
.28708 
.28736 

.95824 
.95816 
•95807 
•95799 
•95791 
•95782 

•30265 
.30292 
.30320 
•30348 
.30376 
.30403 

•95310 
•95301 

•95293 
.95284 

•95275 
.95266 

•31923 
•3J951 
•31979 
.32006 

•32034 
.32061 

.94768 
•94758 
•94749 
•94740 
•94730 
•94721 

•33573 
•33600 
.33627 

•33655 
•33682 
•33710 

.94196 
.94186 
.94176 
.94167 

•94157 
.94147 

23 

22 
21 
20 

11 

43 
44 
45 

i  46 

i:i 

.27068 
.27116 
.27144 
.27172 
.27200 
.27228 

.96261 

•96253 
.96246 

•96238 
.96230 
.96222 

.287^4 
.28792 
28820 
.2^847 
•28875 
.20905 

•95774 
•95766 
-95757 
•95749 
•95740 
•95732 

•30431 

:£$ 

•30514 
•30542 
•30570 

•95257 
.95248 
•95240 
•95231 
.95222 

•95213 

.32089 
.32116 
.32144 
.32171 

•32199 

.32227 

.94712 
.94702 
.94603 
.9461)4 
.94674 
.CJ46CS 

•33737 
•33764 
•33792 
•33819 
•33846 
•33874 

•94137 
.94127 
.94118 
.94108 
.94098 
.94088 

!! 

15 
14 
13 

12 

49 
50 
51 
52 
53 
54 

.27256 
.27284 
.27312 
.27340 
•27368 
•27396 

.96214 
.96206 
.96198 
.96190 
.96182 
•96174 

•28931 
.28959 
.28987 
•29015 
.29042 
.29070 

•95724 
•95715 

.95698 
.95690 
•95681 

•30597 
.30625 

•30653 
.30680 
.30708 
•30736 

.95204 

•95*95 
.95186 

•95177 
.95168 

•95159 

•32254 
.32282 

•32309 
•32337 
•32364 
•32392 

•94656 
.94646 

•94637 
.94627 
.94618 
.94609 

•33901 
•33929 
•33956 
•33983 
.34011 

•34038 

.94078 
.94068 
.94058 
.94049 

•94039 
.94029 

II 
10 

I 

i 
g 

.27424 
.27452 
.27480 
.27508 
•27536 
•27564 

.96166 
.96158 
.96150 
.96142 
.96134 
.96126 

.29098 
.29126 
.29154 
.29182 
.29209 
.29237 

.95673 
.95664 

•95656 
•95647 
•95639 
•95630 

N.  sine 

•30763 
.30791 
•30819 
.30846 
.30874 
.30902 

N.  cos. 

•95150 
•95142 
-95*33 

•95124 

•95^5 
.95106 

.32419 
•32447 
•32474 
•32502 
.32529 
•32557 

•94599 
.94590 
.94580 
•94571 
•9456i 
•94552 

•34065 

•34093 
•34120 

•34H7 
•34175 
.34202 

.94019 
.94009 

•93999 
.93989 

•93979 
.93969 

5 
4 
3 

2 
I 
O 

N.  cos. 

N.  sine 

N.  cos. 

N.  sine 

N.  cos. 

N.  sine 

N.  cos. 

N.  sine 

9 

74° 

73° 

72° 

71° 

70° 

8o 


TABLE  V. 


2O° 

21° 

22° 

23° 

24° 

/ 

N.  sine 

N.  cos. 

N.  sine 

N.  cos. 

N.  sine 

N.  cos. 

N.  sine 

N.  cos. 

N.  sine 

N.  cos. 

60 
59 
58y 

1 

55 
54 

o 
I 

2 

3 

i 

.34202 
.34229 

.34257 
.34284 

•543" 
•34339 
•34366 

•93969 
•93959 
•93949 
•93939 
•93929 
•939J9 
•93909 

•35837 
.35864 
•35891 
•35918 
•35945 
•35973 
.36000 

•93358 
•93348 
•93337 
•93327 
•93316 
•93306 
•93295 

.37461 
•37488 

•37515 
•37542 
.37569 
•37595 
.37622 

•37649 
.37676 

•37703 
•37730 
•37757 
•37784 

.92718 
.92707 
.92697 
.92686 

.92675 
.92664 

•92653 

•39073 
.39100 
.39127 

•39153 
.39180 
.39207 
•39234 

.92050 
.92039 
.92028 
.92016 
.92005 
.91994 
.91982 

.40674 
.40700 
.40727 

•40753 
.40780 
.40806 
•40833 
.40860 
.40886 
.40913 
.40939 

.40992 

•91355 
•91343 
•91331 
•913*9 
•91307 
.91295 
.91283 

i 

9 

10 

ii 

12 

•34393 
.34421 

•34448 
•34475 
•345°3 
•34530 

.93899 
.93889 
.93879 
•93869 
•93859 
.93849 

.36027 

•36054 
.36081 
.36108 

•36135 
.36162 

•93285 
•93274 
.93264 

•93253 
•93243 
•93232 

.92642 
.92631 
.92620 
.92609 
•92598 
•92587 

.39260 
.39287 
•39314 
•39341 
•39367 
•39394 

.91971 

•9*959 
.91948 
.91936 
.91925 
.91914 

.91272 
.91260 
.91248 
.91236 
.91224 
.91212 

53 
52 
5i 
50 
49 
48 

13 
H 

!i 
[I 

•34557 
•34584 
.34612 

•34639 
.34666 

•34694 

•93839 
.93829 

•93819 
.93809 

•93799 

.36190 
.36217 

•36244 
.36271 
.36298 
•36325 

.93222 
.93211 
.93201 
.93190 
.93180 
.93169 

.37811 
•37838 
•37865 
•37892 
•37919 
•37946 

.92576 
•92565 
•92554 
•92543 
•92532 
.92521 

.39421 
.39448 
•39474 
•39501 
•39528 
•39555 

.91902 

.91879 
.91868 
.91856 
.91845 

.41019 
.41045 
.41072 
.41098 
.41125 
.41151 

.91200 
.91188 
.91176 
.97164 
.91152 
.91140 

47 
46 

45 
44 
43 
42 

19 
20 

21 
22 

23 

24 

•34721 
•34748 

•34803 
•34830 
•34857 

•93779 
•93769 
•93759 
•93748 
•93738 
.93728 

•36352 
•3637? 
.36406 

•36434- 
.36461 
.36488 

•93159 
.93148 

•93137 
•93127 
.93116 
93106 

•37973 
•37999 
.38026 

•38*053 
.38080 
.38107 

.92510 

•92499 
.92488 

•92477 
.92466 

•92455 

•3958i 
.39608 

19688 
•39715 

•91833 
.91822 
.91810 
.91799 
.91787 
•9J775 

.41178 
.41204 
.41231 

.41257 
.41284 
.41310 

.91128 
.91116 

.91104 
.91092 
.91080 
.91068 

41 
40 

fs 

11 

2 
2 

29 
30 

.34884 
.34912 

•34939 
.34966 

•34993 
•35021 

•93718 
•93708 
.93698 
•93688 
.93677 
•93667 

•36515 
•36542 
•36569 
.36596 
•36623 
.36650 

•93095 
.93084 

•93074 
•93063 
•93052 
•93042 

•38134 
.38161 
.38188 
.38215 
.38241 
.38268 

•92444 
•92432 
.92421 
.92410 
.92399 
.92388 

•39741 
39768 

39795 
39822 
.39848 
•39875 

.91764 

•91752 
.91741 
.91729 
.91718 
.91706 

•41337 
•41363 
.41390 
.41416 

•41443 
.41469 

.91056 
.91044 
.91032 
.91020 
.91008 
.90996 

35 
34 
33 
32 
3i 
30 

3i 
32 
33 

34 

8 

•35048 

«35075 
•35102 

•35130 
•35157 
•35184 

•93657 
•93647 
.93637 
.93626 
.93616 
.93606 

.36677 
•36704 
.36731 
.36758 
•36785 
.36812 

•93031 
.93020 
.93010 

.92978 

.38295 
•38322 
.3f349 
•38376 
•38403 
•38430 

.92377 
.92366 
•92355 
•92343 
92332 
•92321 

.39902 
.39928 

•39955 
.39982 
.40008 
•40035 

.91694 
.91683 
.91671 
.91660 
.91648 
91636 

.41496 
.41522 
•41549 
.41575 
.41602 
.41628 

.90984 
.90972 
.90960 
.90948 
.90936 
.90924 

27 
26 

25 

24 

23" 

22 
21 
2O 
19 

P 

39 
40 

4i 
42 

•35211 
.35239 
.35266 

.35293 
•35320 
•35347 

•93596 
•93585 
•93575 
•93565 
•93555 
•93544 

•36839 
.36867 

.36894 
•36921 
.36948 

•36975 

.92967 
•92956 
•92945 
•92935 
.92924 
.92913 

•38456 
•38483 
.38510 

.38537 
.38564 
•38591 

.92310 
.92299 
.92287 
.92276 
.92265 
•92254 

.40062 
.40088 
.40115 
.40141 
.40168 
.40195 

.91625 
.91613 
.91601 
.91590 
•91578 
.91566 

•41655 
.41681 
.41707 

•41734 
.41760 
.41787 

!9o875 
.90863 
•90851 

43 
44 

g 

48 

•35375 
•35402 

•35429 
•35456 
•35484 
•355" 

•93534 
•93524 
•935'4 
•93503 
•93493 
•93483 

.37002 
.37029 
37056 
•37083 
.37110 
-37137 

.92902 
.92892 
.92881 
.92870 
.92859 
.92849 

.38617 

.38644 
.38671 
•38698 
.38725 
•38752 

•92243 
.92231 
.92220 
.92209 
.92198 
.92186 

.40221 
.40248 
•40275 
•40301 
.40328 

.40355 

•91555 
•91543 
•9i53i 
•9I5J9 
.91508 
.91496 

.41813 
.41840 
.41866 
.41892 
.41919 
•41945 

•90839 
.90826 
.90814 
.90802 
.90790 
.90778 

!Z 

15 
H 
13 

12 

~II~~ 
10 

I 

49 
50 
51 
52 
53 
S4 

•35538 
•35565 
•35592 
•356i9 
.35647 
•35674 

•93472 
•93462 
•93452 
•93441 
•93431 
.93420 

•37164 
•37I9I 
.37218 

.37245 
.37272 

.37299 

.92838 
.92827 
.92816 
.92805 
.92794 
.92784 

.38778 
.38805 
•38832 

•38859 
.38856 
.38912 

•92175 
.92164 
.92152 
.92141 
.92130 
.92119 

.40381 
.40408 
.40434 
.40461 
.40488 
.40514 

.91484 
.91472 
.91461 
.91449 

•9H37 
.91425 

.41972 
.41998 
.42024 
.42051 
.42077 
.42104 

.90766 

•90753 
.90741 
.90729 
.90717 
.90704 

ft 

5587 
S 

•35701 
.35728 

•35755 
.35782 
•35810 
•35837 

.93410 
.93400 
•93389 
•93379 
•93368 
•93358 

.37326 
•37353 
•37380 
•37407 
•37434 
.37461 

•92773 
.92762 

•92751 
.92740 
.92729 
.92718 

•38939 
.38966 

•38993 
.39020 

•39046 
•39073 

.92107 
.92096 
.92085 
.92073 
.92062 
.92050 

.40541 
.40567 
.40594 
.40621 
.40647 
.40674 

.91414 
.91402 
.91390 
•91378 
.91366 

•91355 

.42130 
.42156 
.42183 
.42205 

•42235 
.42262 

!  90668 
•90655 
•90643 
.90631 

5 
4 
3 

• 
I 

0 



N.  cos. 

N.  sine 

N.  cos. 

N.  sine 

N.  cos. 

N.  sine 

N.  cos. 

N.  sine 

N.  cos. 

N.  sine 

f 

69° 

6§'° 

67° 

66° 

65° 

NATURAL  SINES  AND  COSINES. 


8l 


Vw 

2, 

5° 

2< 

J° 

2' 

r° 

21 

*° 

21 

>° 

/ 

^.  sine 

\.  cos. 

X.  sine 

N.  cos. 

\.  sine 

\.  cos. 

NT.  sine 

^.  cos. 

N.  sine 

N.  cos. 

o 
I 

2 

3 
4 

I 

.42262 
.42288 

•42315 
.42341 
.42367 
.42394 

.42420 

.90631 
.90618 
.90606 

•90594 
.90582 

.90569 
•90557 

•43837 
•43863 
•43889 
.43916 
•43942 
.43968 
•43994 

.89879 
.89867 
.89854 
.89841 
.89828 
.89816 
.89803 

45399 
•45425 
•45451 
•45477 
•45503 
•45529 
•45554 

.89101 

.89087 
.89074 
.89061 
.89048 
•89035 

.89021 

.46947 
•46973 
•46999 
.47024 
.47050 
.47076 
.47101 

.88295 
.88281 
.88267 
.88254 
.88240 
.88226 
.88213 

.48481 
.48506 
•48532 
•48557 
•48583 
.48608 
.48634 

.87462 
.87448 

•87434 
.87420 
.87406 
.87391 
.87377 

60 
59 
5* 
57 
5° 
55 

54 

I 

9 

10 

ii 

12 

.42446 

•42473 
.42499 

•42525 
•42552 
.42578 

•90545 
•90532 
.90520 
.90507 
.90495 
.90483 

.44020 
.44046 
.44072 
.44098 
.44124 
•44151 

.89790 

•89777 
.89764 

•89752 
•89739 
.89726 

.4558o 
.45606 
•45632 
•45658 
.45684 
•45710 

^88968 

•88955 
.88942 

.47127 

•47153 
.47178 
.47204 
.47229 
•47255 

.88199 
.88185 
.88172 
.88158 
.88144 
.88130 

•48659 
.48684 
.48710 

.48735 
.48761 
.48786 

•87363 
•87349 
•87335 
.87321 
.87306 
.87292 

53 
52 
5i 
50 
49 
48 

13 

H 

!i 
jl 

.42604 
.42631 
.42657 
•42683 
.42709 
•42736 

.90470 
.90458 
.90446 

•90433 
.90421 

.90408 

•44177 
.44203 
.44229 

•44255 
.44281 

•44307 

•89713 
.89700 
.89687 

.89674 
.89662 
.89649 

•45736 
.45762 

•45787 
•45813 
45839 
45865 

.88928 
•88915 
.88902 

.88888 
•88875 
.88862 

.47281 
•47306 
•47332 
•47358 
•473»3 
.47409 

.88117 
.88103 
.88089 

.88075 
.88062 
.88048 

.48811 

•48837 
.48862 
.48888 
.48913 
•48938 

.87278 
.87264 
.87250 
•87235 
.87221 
.87207 

47 
46 

45 
44 
43 
42 

19 

20 
21 
22 

23 

24 

.42762 
.42788 

•42815 
.42841 
.42867 
.42894 

.90396 

•90383 
.90371 

•90358 
.90346 

•90334 

•44333 
•44359 
•44385 
.44411 

•44437 
•44464 

.89636 
.89623 
.89610 

•89597 
.89584 

•89571 

.45891 
•45917 
.45942 
.45968 

•45994 
.46020 

.88848 
.88835 
.88822 
.88808 
.88795 
.88782 

•47434 
.47460 
.47486 

•475  » 
•47537 
.47562 

.88034 
.88020 
.88006 
•87993 

•87979 

.87965 

.48964 
.48989 
.49014 
.49040 
.49065 
.49090 

•87193 
.87178 
.87164 
•87150 
.87136 
.87121 

41 
40 

P 

M 

2 

3 

29 
30 

.42920 
.42946 
•42972 
.42999 
•43025 
•43051 

.90321 
.90309 
.90296 

.90284 

.90271 
.90259 

.44490 
.44516 
.44542 
.44568 

•44594 
.44620 

.89558 
•89545 
•89532 
89519 
.89506 
.89493 

.46046 
.46072 
.46097 
.46123 
.46149 
•46i75 

.88768 

•88755 
.88741 
.88728 
•88715 
.88701 

•47588 
•476i4 
.47639 
•47665 
.47690 
.47716 

•8795! 
•87937 
.87923 
.87909 
.87896 

.87882 

.49116 
.49141 
.49166 
.49192 
.49217 
.49242 

.87107 
.87093 
.87079 
.87064 
.87050 
.87036 

35  , 
34 
33 
32 
3i 
30 

31 
32 
33 
34 

35 
36 

•43°77 
.43104 

•43130 
•43156 
.43182 
.43209 

.90246 

.90233 

.90221 
.90208 

.90196 
.90183 

.44646 
.44672 
.44698 
•44724 
•44750 
.44776 

.89480 
.89467 
.89454 
.89441 
.89428 
.89415 

.46201 
.46226 
.46252 
.46278 
.46304 
•46330 

.88688 
.88674 
.88661 
.88647 
.88634 
.88620 

•47741 
•47767 
•47793 
.47818 

•47844 
.47869 

.87868 
.87854 

.87840 
.87826 
.87812 

.87798 

.49268 
.49293 
.49318 
•49344 
•49369 
•49394 

.87021 
.87007 
.86993 
.86978 
.86964 
.86949 

i 

27 
26 

25 
24 

11 

39 
40 
4i 
42 

•43235 
.43261 
.43287 

•43313 
•43340 
•43366 

.90171 
.90158 
.90146 
.90133 

.90120 
.90108 

.44802 
.44828 
.44854 
.44880 
.44906 
.44932 

.89402 
.89389 
•89376 
•89363 
•89350 
89337 

•46355 
.46381 
.46407 
.46433 
•46458 
.46484 

.88607 
.88593 
.88580 
.88566 
•88553 
•88539 

•47895 
.47920 
.47946 
•47971 
•47997 
.48022 

.87784 
.87770 
.87756 
•87743 
.87729 

•87715 

.49419 

•49445 
.49470 

•49495 
.49521 
.49546 

•86935 
.86921 

!86878 
.86863 

23 

22 
21 
20 
19 

43 
44 
45 
46 

47 
1  48 

•43392 
.43418 

•43445 
•43471 
•43497 
•43523 

•90095 
.90082 
.90070 
.90057 
.90045 
.90032 

.44958 
.44984 
45010 

•45036 
.45062 
.45088 

•89324 
.89311 
.89298 
.89285 
.89272 
.89259 

.46510 
•46536 
.46561 
.46587 
.46613 
•46639 

.88526 
.88512 
.88499 
.88485 
.88472 
.88458 

.48048 
.48073 
.48099 
.48124 
.48150 
•48i75 

.87701 
.87687 

•87673 
.87659 

.87645 

.87631 

•49571 
.49596 
.49622 
.49647 
.49672 
•49697 

.86849 
.86834 
.86820 
.86805 
.86791 
•86777 

|| 

*5 
14 

*3 

12 

49 
50 
51 

52 
53 
54 

•43549 
43575 
.43002 
•43628 

43654 
.43680 

.90019 
.90007 

.89994 
.89981 

.89968 
.89956 

•45  "4 
.45140 
.45166 
.45192 
.45218 
•45243 

•89245 
.89232 
.89219 
.89206 
.89193 
.89180 

.46664 
.46690 
.46716 
.46742 
.46767 
•46793 

.88445 
.88431 
.88417 
.88404 
.88390 
•88377 

.48201 
.48226 
.48252 
.48277 

•48303 
.48328 

.87617 
.87603 

.87589 
•87575 
.87561 
.87546 

•49723 
•49748 

•49773 
.49798 
.49824 
.49849 

.86762 
.86748 

•86733 
.86719 
.86704 
.86690 

II 
10 

7 

58 
60 

.43706 
•43733 
•43759 
•43785 
•438" 
.43837 

.89943 
.89930 
.89918 

•89879 

.45269 

.45295 
•45321 
•45347 
•45373 
•45399 

.89167 

.89153 
.89140 
.89127 
.89114 
.89101 

.46819 
.46844 
.46870 
.46896 
.46921 
•46947 

.88363 
.88349 
•88336 
.88322 
.88308 
.88295 

.48354 
.48379 
.48405 
.48430 
.48456 
.48481 

•87532 
.87518 
•87504 
.87490 
.87476 
.87462 

.49874 
.49899 
.49924 
.49950 
•49975 
.50000 

.86675 
.86661 
.86646 
.86632 
.86617 
.86603 

5 
4 
3 

2 
I 
O 

N.  cos 

N.  sine 

N.  cos 

N.  sine 

N.  cos. 

N.  sine 

N.  cos 

N.  sine 

N.  cos. 

N.sine 

/ 

1 

€ 

4° 

6 

3° 

6 

2° 

e 

1° 

€ 

0° 

TABLE  V. 


30° 

31° 

32° 

33° 

34° 

f 

N.  sine 

N.  cos. 

N.  sine 

N.  cos. 

N.  sine 

N.  cos. 

N.  sine 

N.  cos. 

N.  sine 

N.  cos. 

0 

I 

2 

3 
4 

| 

.50000 
•50025 
.50050 
.50076 
.50101 
.50126 
.50151 

.86603 
.86588 
•86573 
•86559 
.86544 
.86530 
.86515 

•51504 
•51529 
•51554 
•51579 
.51604 
.51628 
•5l653 

.85717 
.85702 
.85687 
.85672 

•85657 
.85642 
-85627 

.52992 
•53017 
•53041 
.53066 

•53091 
•53H5 
•53  HO 

.84805 
.84789 
.84774 
•84759 
•84743 
.84728 
.84712 

•54464 
.54488 
•54513 
•54537 
.54561 
•54586 
.54610 

.83867 
•83851 
•83835 
•83819 
.83804 
•83788 
.83772 

.55919 

•55943 
•55968 
•55992 
.56016 
.56040 
.56064 

^82871 
.82855 
.82839 
.82822 
.82806 

60 

II 
H 

55 
54 

2 

9 

10 

ii 

12 

•50176 
.50201 
.50227 
.50252 
.50277 
•50302 

.86501 
.86486 
.86471 

.86457 
.86442 
.86427 

.51678 

•STO 
.51728 

•51753 

« 

.85612 
•85597 
•85582 
•85567 
•85551 
•85536 

•53164 
•53189 
•53214 
•53238 
•53263 
•5328S 

.84697 
.84681 
.84666 
.84650 

•84635 
.84619 

•54635 
•54659 
•54683 
.54708 
•54732 
•54756 

.83750 
.83740 
.83724 
.83708 
.83692 
.83676 

.56088 
.56112 

•56136 
.56160 
.56184 
.56208 

.82790 
•82773 
.82757 
.82741 
.82724 
.82708 

53 
52 
5i 
50 
49 
48 

13 
14 

is 
[I 

•50327 
•50352 
•50377 
•50403 
.50428 

•5°453 

.86413 
.86398 
.86384 
.86369 
•86354 
.86340 

.51828 
.51852 

•51877 
.51902 

•51927 
•5!952 

•85521 
.85506 
.85491 
.85476 
.85461 
.85446 

•53312 
•53337 
•5336i 
•53386 

-534" 

•53435 

.84604 
.84588 
•84573 
•84557 
•84542 
.84526 

•5478i 
•54805 
•54829 
•54854 
.54878 
.54902 

.83660 

•83645 
.83629 

•83613 
•83597 
.83581 

.56232 
.56256 
.56280 
•56305 
•56329 
.56353 

.82692 
.82675 
.82659 
.82643 
.82626 
.82610 

47 
46 

45 
44 
43 
42 

19 

20 
21 
22 
23 

24 

.50478 
•50503 
•50528 

•50553 
.50578 
.50603 

.86325 
.86310 

!  86266 
.86251 

•51977 
.52002 
.52026 

•52051 
.52076 
.52101 

•85431 
•85416 

.85401 
•85385 
•85370 
•85355 

•53460 
.53484 
•53509 
•53534 
•53558 
•53583 

.84511 
•84495 
.84480 
.84464 
.84448 
•84433 

•54927 
•54951 
•54975 
•54999 
•55024 
•55048 

•83565 
•83549 
•83533 
•83517 
•83501 
•83485 

•56377 
.56401 

•56425 
•56449 
•56473 
•56497 

•82593 
•82577 
.82561 
•82544 
.82528 
.82511 

41 
40 

1 

II 

3 

29 

30 

.50628 

•50654 
.50679 
.50704 
.50729 
.50754 

86237 
.86222 
.86207 
.86192 
.86178 
.86163 

.52126 
•52151 
•52175 
.52200 
.52225 
•52250 

•85340 

•85325 
.85310 
.85294 
.85279 
.85264 

•53607 
•53632 
•53656 
.53681 
•53705 
•5373° 

.84417 
.84402 
.84386 
.84370 
.84355 
•84339 

•55072 
•55097 
•55I2I 
•55145 
•55169 
•55194 

.83469 
•83453 
•83437 
.83421 

•83405 
•83389 

•56521 
•56545 
•56569 
•56593 
•56617 
.56641 

.82495 
.82478 
.82462 
.82446 
.82429 
.82413 

35 
34 
33 
32 
31 
30 

'11 

11 

25 

24 

3i 
32 
33 

34 

ii 

.50779 
.50804 
.50829 

•50854 
.50879 
.50904 

.86148 
•86133 
.86119 
.86104 
.86089 
.86074 

•52275 
•52299 
•52324 
•52349 
•52374 
•52399 

.85249 

•85234 
.85218 
.85203 
.85188 
•85173 

•53754 
•53779 
•53804 
.53828 

•53853 
•53877 

.84324 
.84308 
.84292 
.84277 
.84261 

•84245 

.55218 
•55242 
•55266 
•55291 
•55315 
•55339 

•83373 
•83356 
.83340 
•83324 
.83308 
.83292 

.56665 
.56689 

•56736 
.56760 
.56784 

.82396 
.82380 
•82363 

•82347 
•82330 
•82314 

11 

39 
40 

4i 
42 

.50929 
•50954 
•50979 
.51004 
.51029 
•51054 

.86059 
.86045 
.86030 
.86015 
.86000 
•85985 

•52423 
.52448 

•52473 
.52498 
.52522 
•52547 

.85157 
•85142 
.85127 
.85112 
.85096 
.85081 

•53902 
•53926 
•53951 
•53975 
.54000 
.54024 

•84230 
.84214 
.84198 
.84182 
.84167 
.84151 

:P 

•55412 
•55436 
•55460 
•55484 

•83276 
.83260 
.83244 
.83228 
.83212 
•83195 

.56808 
.56832 

'56904 
.56928 

.82297 
.82281 
.82264 
.82248 
.82231 
.82214 

23 

22 
21 
2O 

43 
44 

1 

2 

•51079 
.51104 
.51129 
•5H54 
•5H79 
.51204 

.85970 
•85956 
.85941 
.85926 
.85911 
.85896 

•52572 
•52597 
52621 
.52646 
.52671 
.52696 

.85066 
•85051 

•85035 
.85020 
.85005 
.84989 

.54049 
•54073 
•54097 
.54122 
.54146 
.54171 

•84135 
.84120 
.84104 
.84088 
.84072 
.84057 

•55509 
•55533 
•55557 
•5558i 
•55605 
•55630 

•83179 
•83163 
•83147 
•83131 
.83115 
.83098 

.56952 
.56976 
.57000 
•57024 
•57047 
•57071 

.82198 
.82181 
.82165 
.82148 
.82132 
.82115 

\l 

15 
H 
'3 
12 

49 
50 
51 
52 
53 
54 

.51229 
•51254 
•51279 
•5I3°4 
•5I329 
:5i354 

.85881 
.85866 
.85851 
.85836 
.85821 
.85806 

.52720 
•52745 
•52770 
.52794 
.52819 
•52844 

•84974 
.84959 
.84943 
.84928 
.84913 
.84897 

•54195 
.54220 

•54244 
.54269 

•54293 
•54317 

.84041 
.84025 
.84009 
•83994 
.83978 
.83962 

•55654 
•55678 
•55702 
•55726 
•55750 
•55775 

.83082 
.83066 
.83050 
•83034 
•83017 
.83001 

•57095 
•57"9 
•57143 
.57167 
.57191 
•57215 

.82098 
.82082 
.82065 
.82048 
.82032 
.82015 

II 
10 

| 

P 

R 

e 

•5I379 
.51404 
.51429 
•51454 
•S«479 
•51504 

.85792 

•85777 
.85762 
•85747 
•85732 
•85717 

.52869 

•52893 
.52918 

•52943 
•52967 
.52992 

.84882 
.84866 
.84851 
.84836 
.84820 
.84805 

•54342 
.54366 
•54391 
.54415 
.54440 
.54464 

.83946 
•83930 

•83915 
.83899 
.83883 
•83867 

•55799 
•55823 
.55847 
•55871 
•55895 
•55919 

.82985 
.82969 

•82953 
.82936 
.82920 
.82904 

•5723* 
.57262 
.57286 
•57310 
.57334 
•57358 

.81999 
.81982 

•81965 
.81949 
.81932 
•81915 

5 
4 
3 

2 
I 
O 

N.  cos. 

N.  sine 

N.  cos. 

N.  sine 

N.  cos. 

N.  sine 

N.  cos. 

N.  sine 

N.  cos. 

N.  sine 

f 

59° 

.  68° 

57° 

56° 

55° 

NATURAL  SINES  AND  COSINES.               8 

35° 

36° 

37° 

38° 

39° 

1 

f 

N.  sine 

N.  cos. 

N.  sine 

N.  cos. 

N.  sine 

N.  cos. 

N.  sine 

N.  cos. 

N.  sine 

N.  cos. 

0 

i 

2 

3 
4 

I 

•57358 
•573^1 
•57405 
•57429 
•57453 
•57477 
•57501 

•81915 
.81899 
.81882 
.81865 
.81848 
.81832 
.81815 

.58779 
.58802 
.58826 
.58849 

•58873 
.58896 
.58920 

.80902 
.80885 
.80867 
.80850 
.80833 
.80816 
.80799 

.60182 
.60205 
.60228 
.60251 
.60274 
.60298 
.60321 

.79864 
.79846 
.79829 
.79811 
•79793 
•79776 
•79758 

.61566 
•61589 
.61612 
•61635 
.61658 
.61681 
.61704 

.78801 
•78783 
•78765 
.78747 
.78729 
.78711 
.78694 

.62932 

•62955 
.62977 
.63000 
.63022 
•63045 
.63068 

.77715 
.77696 
•77678 
.77660 
.77641 
.77623 
.77605 

60 

* 

H 

55  ' 
54  I 

I 

9 

1C 

ii 

12 

13 

14 

3 

17 
18 

•57524 
•57548 
•57572 
•57596 
•57619 
•57643 

.81798 
.81782 
.81765 
.81748 
.81731 
.81714 

•58943 
•58967 
.58990 
.59014 

•59037 
.59061 

.80782 
.80765 
.80748 
.80730 
.80713 
.80696 

.60344 
.60367 
.60390 
.60414 
.60437 
.60460 

•79741 
•79723 
.79706 
.79688 
.79671 
•79653 

.61726 

•6i749 
.61772 
.61795 

!6i84i 

.78676 
.78658 
.78640 
.78622 
.78604 
.78586 

.63090 
•63113 

I3!3! 

.63180 
•63203 

•77586 
•77568 
•77550 
•77531 
•77513 
•77494 

53 
52 
5i 
So 
49 
48 

.57667 
•57691 
•57715 
•57738 
•57762 
•57786 

.81698 
.81681 
.81664 
.81647 
.81631 
.81614 

.59084 
.59108 
•59I3I 
•59154 
.59178 
.59201 

.80679 
.80662 
.80644 
.80627 
.80610 
•80593 

.60483 
.60506 
•60529 

•60553 
.60576 
.60599 

.79635 
.79618 
.79600 
.79583 
•79565 
•79547 

.61864 
.61887 
.61909 
.61932 

.61955 
.61978 

.  78568 
•78550 
•78532 
.785H 
.78496 
.78478 

•63225 
.63248 
.63271 
•63293 
•63316 
•63338 

.77476 
•77458 

•77439 
.77421 
.77402 
•77384 

47 
46 

45 
44 
43 
42 

19 

20 
21 
22 

23 

24 

•57810 
•57833 
•57857 
.57881 

•57904 
.57928 

igg 

.81563 
.81546 
.81530 
•81513 

•59225 
.59248 
.59272 

•59295 
•59318 
•59342 

.80576 
.80558 
.80541 
.80524 
.80507 
.80489 

.60622 
.60645 
.60668 
.60691 
.60714 
•60738 

•79530 
•79512 
•79494 
•79477 
•79459 
.79441 

.62001 
.62024 
.62046 
.62069 
.62092 
.62115 

.78460 
.78442 
.78424 
.78405 

•78387 
.78369 

.63361 

•*3382 

.63406 

.63428 
•63451 
•63473 

.77366 
•77347 
•77329 
.77.310 
.77292 
.77273 

41 
40 

I 

3 
2 

29 
30 

•57952 
.57976 

•57999 
-58023 
.58047 
.58070 

.81496 

.81479 
.81462 
.81445 
.81428 
.81412 

•59365 
•59389 
.59412 

.59436 
-59459 
.59482 

.80472 
•80*455 
•80438 
.80420 
.80403 
.80386 

.60761 
.60784 
.60807 
.60830 
.60853 
.60876 

.79424 
.79406 
.79388 
•79371 
•79353 
•79335 

.62138 
.62160 
.62183 
.62206 
.62229 
.62251 

•78351 
•78333 
•78315 
.78297 
.78279 
.78261 

.63496 
.63518 
•63540 
•63563 
•63585 
.63608 

•77255 
•77236 
.77218 
.77199 
.77181 
.77162 

35 
34 
33 
32 
3i 
30 

31 
32 
33 
34 

g 

.58094 
.58118 
.58141 
.58165 
.58189 
.58212 

.81395 
•81378 
.81361 
.81344 
•81327 
.81310 

•59506 
-59529 
•59552 
•59576 
•59599 
.59622 

.80368 
.80351 
•80334 
.80316 
.80299 
.80282 

.60899 
.60922 
.60945 
.60968 
.60991 
.61015 

•793l8 
.79300 
.79282 
.79264 
.79247 
.79229 

.62274 
.62297 
.62320 
.62342 
•62365 
.62388 

.78243 
.78225 
.78206 
.78188 
.78170 
•78152 

.63630 
.63653 
•63675 
.63698 
.63720 
63742 

•77144 
•77125 
.77107 
.77088 
.77070 
•77051 

2 
25 

24 

S 

39 
40 

4i 

42 

•58236 
.58260 
.58283 
•58307 
•58330 
•58354 

•81293 
.81276 
.81259 
.81242 
.81225 
.81208 

•59646 
.59669 

•59693 
•59716 
•59739 
•59/63 

.80264 
.80247 
.80230 
.80212 
.80195 
.80178 

.61038 
.61061 
.61084 
.61107 
.61130 
•6"53 

.79211 

.79193 
.79176 

•79158 
.79140 
.79122 

.62411 

•62433 
.62456 
.62479 
.62502 
.62524 

•78i34 
.78116 
.78098 
.78079 
.78061 
.78043 

•63765 
.63787 
.63810 
.63832 
.63854 
.63877 

•77033 
.77014 
.76996 
.76977 

•76959 
.76940 

23 

22 
21 
2O 

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15 

H 
13 

12 

43 

44 
45 
46 
47 

48 

•58378 
.58401 

•58425 
.58449 
.58472 
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.81191 
.81174 

•8n57 
.81140 
.81123 
.81106 

•59786 
.59809 
59832 
•59856 
•598/9 
.59902 

.80160 
.80143 
.80125 
.80108 
.80091 
.80073 

.61176 
.61199 
.61222 
•61245 
.61268 
.61291 

•79105 
•79087 
.79069 
•79051 
•79033 
.79016 

.62547 
.62570 
.62592 
.62615 
.62638 
.62660 

.78025 
.  78007 
•77988 
•779/0 
•77952 
•77934 

.63899 
.63922 
.63944 
.63966 
.63989 
.64011 

-.76921 
.76903 
.76884 
.76866 
.76847 
.76828 

49 
50 
51 
52 
53 
54 

.58519 
•58543 
•58567 
•58590 
.58614 
•58637 

.81089 
.81072 
•81055 
.81038 
.81021 
.81004 

.59926 
•59949 
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.60019 
.60042 

.80056 
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.80021 
.80003 
.79986 
.79968 

.61314 

:£$ 

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.61406 
.61429 

.78998 
.78980 
.78962 
.78944 
.78926 
.78908 

.62683 
.62706 
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.62751 
•62774 
.62796 

.77916 

•77897 
.77879 
.77861 

•77843 
.77824 

•64033 
.64056 
.64078 
.64100 
.64123 
.64145 

.76810 
.76791 
.76772 
•76754 
•76735 
.76717 

II 
IO 

I 

55 
56 

8 

§ 

.58661 
.58684 
•58708 
•58731 
.58755 
.58779 

.80987 
.80970 
•80953 
.80936 
.80919 
.80902 

.60065 
.60089 
.60112 
•60135 
.60158 
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•79951 
•79934 
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.61451 
•61474 
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.78891 
•78873 
•78855 
•78837 
.78819 
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.62819 
.62842 
.62864 
.62887 
.62909 
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.77806 
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.64167 
.64190 
.64212 
.64234 
.64256 
.64279 

.76698 

•76679 
.76661 
.76642 
.76623 
.76604 

N.  sine 

S 
4 

3 

2 

I 
O 

N.  cos. 

N.  sine 

N.  cos. 

N.  sine 

N.  cos. 

N.  sine 

N.  cos. 

N.  sine 

N.  cos. 

i 

54° 

53° 

52° 

51° 

50° 

TABLE  V. 


,  40° 

41° 

42° 

43° 

44° 

9 

N.  sine 

N.  cos. 

N.  sine  N.  cos. 

N.  sine 

N.  cos. 

N.  sine 

N.  cos. 

N.  sine 

N.  cos. 

60 

% 
H  l 

55  . 
54 

O 

I 
2 

3 
4 

I 

.64279 
.64301 

•64323 
.64346 
.64368 
.64390 
.64412 

.  76604 
•76586 
•76567 
.76548 

•76530 
.76511 
.76492 

.65606 
.65628 
.65650 
.65672 
.65694 
•65716 
•65738 

•75471 
•75452 
•75433 
•75414 
•75395 
•75375 
•75356 

66913 
•66935 
.66956 
.66978 
.66999 
.67021 
.67043 

•743H 
•74295 
•74276 
.74256 

•74237 
.74217 
.74198 

.68200 
.68221 
.68242 
.68264 
.68285 
.68306 
.68327 

•73135 
.73116 
•73096 
.73076 
•73056 
•73036 
.73016 

.69466 
.69487 
.69508 
.69529 
.69549 
.69570 
.69591 

•71934 
.71914 
.71894 
•71873 
•71853 
•71833 
.71813 

i 

9 

10 

ii 

12 

•64435 
•64457 
•64479 
.64501 
.64524 
.64546 

•76473 
•76455 
•76436 
.  7641  7 
.  76398 
•  76380 

•65759 
.65781 
•65803 
.65825 
-65847 
.65869 

•75337 
•753i8 

•75299 
.75280 

•75261 
•75241 

.67064 
.67086 
.67107 
.67129 
.67151 
.67172 

.74178 
.74159 
•74139 
.74120 
.74100 
.74080 

.68349 
•68370 
•68391 
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.72996 
.72976 
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.69612 
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'%£& 

.69696 

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.71792 
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53 
52 
5i 
50 
49 
48 

13 
H 

:i 

17 

18 

.64568 
.64590 
.64612 
•64635 
•64657 
.64679 

.76361 
.76342 
.76323 

•76304 
.76286 
.76267 

.65891 
65913 
•65935 
.65956 
.65978 
.66000 

.75222 
•75203 
•75184 
•75165 
•75H6 
.75126 

.67194 
•67215 
.67237 
.67258 
.67280 
.67301 

.74061 
.74041 
.74022 
.74002 
•73983 
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.68476 

.68497 
.68518 
.68539 
.68561 
.68582 

•72877 
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•72837 
.72817 
.72797 
•72777 

.69737 
.69758 
.69779 
.69800 
.69821 
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.71671 
.71650 
.71630 
.71610 
•71590 
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47 
46 
45 
44 
43 
42 

19 

20 
21 
22 
23 

24 

.64701 
.64723 
.64746 
.64768 
.64790 
.64812 

.76248 
.76229 
.76210 
.76192 
•76i73 
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.66022 
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.66066 
.66088 
.66109 
.66131 

•75107 
.75088 
.75069 
•75050 
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•67323 

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•67387 
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•6743° 

•73944 
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.68603 
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.68645 
.68666 
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•72757 
.72737 
.72717 
.72697 
.72677 
.72657 

.69862 
.69883 
.69904 
.69925 
.69946 
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•71549 
•71529 
.71508 
.71488 
.71468 
•71447 

41 
40 

11 

H 

II 

2 

29 
30 

.64834 
.64856 
.64878 
.64901 
.64923 
.64945 

•76135 
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•76078 
•76059 
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•66153 

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.66197 

.66218 
.66240 
.66262 

.74992 
•74973 
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.67452 
•67473 
•67495 
.67516 

•67538 
•67559 

•73826 
.73806 

•73787 
.73767 

•73747 
•73728 

.68730 
.68751 
.68772 

•68793 
.68814 

•68835 

.72637 
.72617 

•72597 
•72577 
•72557 
•72537 

.69987 
.70008 
.70029 
.70049 
.70070 
.70091 

.71427 
.71407 
•71386 
.71366 
•71345 
•71325 

35 
34 
33 
32 
31 
30 

3* 
32 
33 

34 

P 

.64967 

.64989 
.65011 

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.  76022 
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.66284 
.66306 
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.66349 

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.66393 

.74876 
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.74818 

•74799 

.74780 

95 

.67623 
.67645 
.67666 
.67688 

•73708 
.73688 
.73669 
•73649 
•73629 
.73610 

.68857 
.68878 
.68899 
.68920 
.68941 
.68962 

•72517 
•72497 
•72477 
•72457 
•72437 
.72417 

.70112 
.70132 

•70153 
.70174 

•70195 
.70215 

•71305 
.71284 
.71264 

•71243 
.71223 
.71203 

3 

11 
25 
24 

P 

39 
40 

4i 
42 

.65100 
.65122 

3% 

.65188 
.65210 

•75870 
.75851 
•75832 
•758i3 

.66414 
.66436 
.66458 
.66480 
.66501 
•66523 

.74760 

•74741 
.74722 

•74703 
.74683 
.74664 

.67709 
.67730 
.67752 
•67773 
•67795 
.67816 

•73590 
•73570 
•73551 
•73531 
•735" 
•73491 

.68983 
.69004 
.69025 
.69046 
.69067 
.69088 

•72397 
.72377 

•72357 
•72337 
.72317 
.72297 

.70236 
•70257 
.70277 
.70298 
.70319 
•70339 

.71182 
.71162 
.71141 
.71121 
.71100 
.71080 

23 

22 
21 

2O 

lo 

43 
44 

$ 

i 

•65232 
£8 

.65298 
.65320 
•65342 

•75794 
•75775 
•75756 
•75738 
•75719 
.75700 

•66545 
.66566 
66588 
.66610 
.66632 
.66653 

•74644 
.74625 
.74606 
•74586 
•74567 
•74548 

.67837 

•67859 
.67880 
.67901 

•67923 
.67944 

•73472 
•73452 
-73432 
•73413 
•73393 
•73373 

.69109 
.69130 
•69151 
.69172 
.69193 
.69214 

.72277 

•72257 
.72236 
.72216 
.72196 
.72176 

.70360 
.70381 
.70401 
.70422 
•70443 
•70463 

•71059 
.71039 
.71019 
.70998 
.70978 
•70957 

11 

15 

H 
13 

12 

49 
50 
51 
52 
53 
54 

•65364 
•65386 
.65408 
•65430 
.65452 
•65474 

.75680 
.75661 
.75642 
•75623 
.75604 
•75585 

.66675 
.66697 
.66718 
.66740 
.66762 
.66783 

.74528 

•74509 
.74489 
.74470 
•74451 
•74431 

.67965 

.68029 
.68051 
.68072 

•73353 
•73333 
•733!4 
•73294 
•73274 
•73254 

•69235 
.69256 
.69277 
.69298 
.69319 
•69340 

•72156 
.72136 
.72116 
.72095 

•72075 
•72055 

.70484 
•70505 
•70525 
•70546 
•70567 
•70587 

.70937 
.70916 
.70896 

•70875 
•70855 
.70834 

II 
10 

1 

6 

P 
II 
g 

.65496 
.65518 
•65540 
.65562 

l& 

.75566 
•75547 
•75528 
•75509 
•75490 
•75471 

.66805 
.66827 
.66848 
.66870 
.66891 
•66913 

.74412 
•74392 
•74373 
•74353 
•74334 
•743H 

.68093 
.68115 
.68136 
.68157 
.68179 
.68200 

•73234 
.73215 
•73*95 
•73*75 
•73155 
•73135 

•69361 
.69382 

•69403 
.69424 

•69445 
.69466 

•72035 
•72015 
.71995 
.71974 
.71954 
.71934 

.70608 
.70628 
.70649 
.70670 
.70690 
.70711 

•70813 

•70793 
.70772 
.70752 

-70731 
.70711 

5 
4 
3 

2 

I 
0 

N.  cos. 

N.  sine 

N.  cos. 

N.  sine 

N.  cos. 

N.  sine 

N.  cos. 

N.  sine 

N.  cos. 

N.  sine 

9 

49° 

48° 

47° 

46° 

45° 

TABLE  VI.—  ADDITION  AND  SUBTRACTION  LOGARITHMS. 

8 

TABLE  VI. 

;  ADDITION  AND  SUBTRACTION  LOGARITHMS. 

PRECEPTS. 

I.   Wfan  difference  of  given  logarithms  is  less  than  2.OO. 

ADDITION.  —  Enter  table  with   difference  between 

logarithms 

as  Arg.  A,  and  take  out  B. 

Add  B  to  subtracted  logarithm. 

SUBTRACTION.  —  Subtract  lesser  from  greater  logarithm; 

enter 

with  the  difference  as  B,  and  take  out  A. 

Add  A  to  the  subtracted  logarithm. 

II.   When  difference  of  given  logarithms  exceeds  2.OO. 

Subtract  lesser  from  greater. 

ADDITION.  —  Enter  table  with  difference  as  Arg.  At  take  out 

B—A  and  add  it  to  the  greater  logarithm. 

SUBTRACTION.  —  Enter  column  B  with  difference  of 

logarithms  ; 

take  out  B—A,  and  subtract  it  from  greater  logarithm. 

A. 

B.     0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. 

Pts. 

5- 

o.oo  ooo 

001 

OOI 

OOI 

OOI 

OOI 

002 

002 

"003 

"ooj 

m^mmm 

•«••••• 

G.O 

004 

004 

005 

005 

005 

005 

005 

005 

005 

005 

6.1 

005 

006 

006 

006 

006 

006 

006 

006 

007 

007 

3 

4 

5 

6 

6.2 

007 

007 

007 

007 

008 

008 

008 

008 

008 

008 

i 

0.3 

o  4 

0.3 

0.6 

6.3 

009 

009 

009 

009 

OIO 

OIO 

OIO 

OIO 

OIO 

Oil 

2 

0.6 

08 

I.O 

1.3 

3 

0.9 

Z    2 

1.5 

1.8 

6.4 

on 

on 

Oil 

OI2 

OI2 

012 

013 

013 

013 

013 

4 

1.3 

i  6 

3.O 

a  4 

6.5 

014 

014 

014 

015 

015 

015 

016 

016 

017 

017 

5 

i.  5 

3  0 

9*5 

3.0 

6.6 

017 

018 

018 

019 

019 

019 

020 

020 

02  1 

02  1 

6 

1.8 

a  4 

3-0 

3-6 

6.7 

022 

022 

023 

023 

024 

024 

025 

026 

026 

027 

7 
8 

3.1 
8.4 

3  8 

3  2 

3-5 
4.0 

4-a 
4.8 

68 

027 

028 

029 

O2Q 

O3O 

O3I 

Oil 

O32 

o^* 

014 

6.9 

034 

035 

036 

X 

037 

038 

039 

J 

040 

*J 

O4I 

ijj 

041 

JT^ 

042 

7.0 

043 

044 

045 

047 

048 

049 

050 

051 

052 

053 

7-1 

055 

056 

057 

059 

060 

061 

063 

064 

066 

067 

j 

7 

O    7 

0.8 

9 

TO 

7-2 

069 

070 

072 

074 

075 

077 

079 

08  1 

083 

085 

2 

w.y 

1.4 

1.6 

0.9 

1.8 

2.O 

7-3 

087 

089 

091 

093 

095 

097 

099 

102 

104 

106 

3 

2.1 

3-4 

2.7 

3-o 

7-4 

109 

in 

ii/ 

117 

119 

122 

125 

128 

131 

134 

4 

2.8 

3-2 

3-6 

4.0 

7-5 

137 

140 

144 

147 

150 

154 

157 

161 

165 

169 

5 
6 

3-5 
4-2 

4.0 

4.8 

4-5 

5      A 

5-o 
6.0 

7.6 

173 

177 

181 

185 

I89 

194 

198 

203 

207 

212 

7 

4-9 

5-6 

'^ 

6-3 

7.0 

7-7 

217 

222 

227 

233 

238 

244 

249 

255 

261 

267 

8 

5-6 

6-4 

7.2 

8.0 

7-8 

273 

280 

286 

293 

299 

306 

313 

321 

328 

336 

9 

•3 

7.3 

8.1 

9.0 

79 

344 

352 

360 

368 

377 

385 

394 

403 

413 

422 

1    8.0 

432 

442 

452 

463 

474 

485 

496 

507 

519 

531 

1    A> 

B.     0 

1 

2 

a 

.   4 

5 

6 

7 

8 

9 

Prop. 

Pts. 

86 


TABLE  VL 


ADD  i  loS<*  ~  loS*  =  A>          Qrm  /  loS*  ~  loS<*  =  &• 
AmMlog(0  +  £)  =  log0  +  ^.      SU3'Uog(0~£)  =  log£  +  ^. 

A. 

rsloo 

8.01 

8.02 

8.03 

8.04; 
8.05 
8.06 

8.07 
8.08 
8.09 

8.10 
8.  ii 

8.12 

8.13 

8.14 

8.15 
8.16 

8.17 
8.18 
8.19 

8.20 

8.21 
8.22 

1  8-23' 

8.24 
8.25 

8.26 

8.27 
8.28 

8.29 

8.30 

«.3i 
8.32 

I- 

Sip 

l:5 

8.39 
8.40 

Itt 

8.43 

8.44 

I'4! 
8.46 

8.47 
8.48 
8.49 
8.50 

11.  0 
o.oo  432 

1 

433 

i 

434 

a 

435 

4 

~ 

5 

~ 

0 

•••••••M 

438 

7 

m^m^mm 

439 

8 
440 

9 

44i 

"451 
462 

473 

483 
495 
506 

518 
530 

542 

Proi 

^••^•^••w 

I 

2 

3 

i 
i 

9 

i 

2 

3 
4 

i 
I 

9 

i 

2 

3 
4 

1 
I 

9 

).Pt8.    1 
•'•• 

£* 

0.8 

J:5 
\i 

:\ 

°3 
0.6 

?:I 
1:1 

•* 

'.' 

°*1 

0.8    \ 

1.2 

1.6 

2.0 

:i 

!:J 

442 

452 

463 

474 
485 
496 
507 
519 
53i 

443 
453 
464 

475 
486 

497 
508 
520 
532 

444 
454 
465 

476 
487 
498 

510 
521 
533 

445 
456 
466 

477 

488 

499 

5" 
523 
535 

446 

457 
467 

478 
489 
500 

512 
524 
536 

447 
458 
468 

479 
490 
502 

513 
525 
537 

448 

459 
469 

480 

49  i 
503 

5H 
526 
538 

449 
460 
470 

481 

492 
504 

515 
527 
540 

450 
461 
47i 
482 
494 
505 

517 
529 
541 

543 

5|6 
569 
582 

595 
609 
623 

638 
652 
667 
683 

545 

546 

547 

548 

550 

55i 

552 

553 

555 

557 
570 

583 

597 
611 
625 

639 
654 
669 

558 
57i 
585 

598 
612 
626 

641 

655 
671 

560 

$ 

599 
613 
628 

642 

657 

672 

561 
574 
587 
601 

6i5 
629 

644 
658 
674 

562 

575 
589 

602 
616 
630 

645 
660 
675 

564 

577 
590 

604 
618 
632 
646 
661 
677 

565 
578 
59i 
605 
619 
633 
648 
663 
678 

566 
579 
593 
606 
620 
635 
649 
664 
680 

567 
58i 
594 
608 
622 
636 

& 

681 

684 

686 

688 

689 

691 

692 

694 

696 

697 

699 

715 
731 

748 
766 

783 
801 
820 
839 
858 

"8^8 
898 
919 

940 
962 
984 
o.oi  006 
030 
Q53 
077 

700 
716 
733 
750 
767 
785 
803 
822 
841 

702 
718 
735 

752 
769 

787 
805 
823 
842 

703 
720 

736 

753 
771 

789 
807 
825 
844 

705 
721 
738 

755 
773 
790 

809 
827 
846 

707 

723 
740 

757 
774 
792 
810 
829 
848 

708 

725 
741 

759 
776 
794 
812 

831 
850 

710 
726 
743 
760 
778 
796 

814 

833 
852 

712 
728 
745 
762 
780 
798 

816 
835 
854 

713 

730 
747 
764 
78i 
799 
818 

837 
856 

860 

862 

864 

866 

868 

870 

872 

874 

876 

880 
900 
921 

942 
964 
986 

009 
032 
056 

882 
902 
923 

944 

966 
988 

on 

034 
058 

884 

904 

925 
946 
968 
99o 

013 

037 
060 

886 
906 
927 

948 
970 

993 
016 

039 
063 

888 
908 
929 

95i 
973 
995 
018 
041 
065 

890 
910 
93i 

953 

975 
997 
020 

044 
068 

892 
912 
933 

955 
977 
999 

022 

046 
070 

894 
915 
936 

957 
979 

*002 

O23 

048 
073 

896 
917 
938 

P? 
*<x>4 
027 

°51 
075 

080 

082 

085 

087 

090 

092 

095 

097 

IOO 

1  02 
128 
153 
1  80 
207 
23 
263 
292 
322 
352 

105 
130 
156 

183 

2IC 

23* 

266 
295 
32] 

107 
133 
159 
185 

21; 

240 

269 
298 

328 

HO 
135 

161 

1  88 
215 
243 
272 
30 
33 

112 
138 
164 

I9I 

218 

246 

275 
304 

334 

H5 

140 

167 
193 

221 
249 
27* 
307 

337 

117 

143 
169 

196 

22; 
252 

280 
310 
340 

120 
146 
172 

199 
226 
25" 
283 
313 

343 

122 
148 
175 
2O2 
229 
257 
286 
316 
346 

125 

151 
177 

204 

232 

260 
289 

319 

349 

35! 

358 

36 

364 

36* 

37 

374 

377 

380 

II  A- 

B.  0 

1 

2 

8 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

o 


ADDITION  AND  SUBTRACTION  LOGARITHMS.          g 

Ann  J  lQ%b  ""  lo%a  —  ^'          STTB  J  loS<*  —  lo§^  =  A 
'  I  log  («  +  *)  =  log*  -f  A     bIJ  i  log  (a  -  J)  =  log*  +  A. 

A. 
~&60 

8.5i 
8.52 

8-53 
8.54 
8-55 
8.56 

8.57 
8.58 

8-59 
8.60 

8.61 
8.62 
8.63 

8.64 
8.65 
8.66 

8.67 
8.68 
8.69 

8.70 

8.7i 
8.72 

8.73 
8.74 
8.75 
8.76 

*.77 
8.78 

8.79 
8.80 
8.81 
8.82 
8.83 

8.84 
8.85 
8.86 

8.87 
8.88 
8.89 

8.90 

8.91 
8.92 
8-93 
8.94 
8.95 
8.96 

8.97 
8.98 
8.99 

9.00 

B.  0 

o.oi  352 

383 
415 
447 
480 
5H 
549 
584 
621 
658 

695 

1 

"355 

2 
IP 

a 

~i 

4 

364 

5 

368 

6 

37i 

7 
374 

8 
377 

9 

380 

P 

I 

2 

3 
4 

i 
I 

9 

I 

2 

3 

i 
i 

9 

i 

2 

3 

I 
I 

9 

i 

2 

3 

i 
I 

9 

rop. 

MHM^ 

s 
0.3 
0.6 
0.9 

1.2 

*:I 

2.1 
2.4 
2.7 

5 

0-5 

I.O 

«-5 

2.O 

2-5 

3-o 
3-5 
4-0 

4-5 

7 

0.7 
1-4 

2.1 

2.8 

3-5 
4.2 

4-9 

I'6 
0-3 

9 

!:! 

11 

4-5 

*1 

*; 

Pts. 

mmm^^m^mmmm 

4 
0.4 

0.8 

1.2 

1.6 

2.O 

2:3 

& 

6 
0.6 

1.2 

1.8 
2.4 
3-o 
3-6 
4.2 
4.8 
54 

8 

0.8 
1.6 
2.4 
3-2 
4.0 
4.8 

I'6 
6.4 

7.2 
so 

I.O 
2.O 
30 
4-0 

5-0 
6.0 
7.0 
8.0 
9.0 

386 
418 
450 

484 
518 
552 
588 
624 
661 

389 
421 
454 

487 
521 
556 

I9l 
628 

665 
"703 

393 
424 
457 
490 
5^5 
559 

595 
632 
669 

396 
428 
460 

494 

528 
563 

599 
635 
673 

399 
43i 
464 

497 
531 
566 

602 

639 
676 

402 
434 
467 
501 

535 
570 

606 

643 
680 

405 
437 
470 

504 
538 
574 
610 
646 
684 

408 
441 
474 
507 
542 
577 
613 
650 
688 

412 
444 
477 

5" 

545 
581 

617 
654 
692 

699 

707 

711 

715 

719 

722 

726 

730 

734 
774 
814 

856 
898 
941 

985 

O.O2  030 
077 

124 

738 
778 
8.'8 

860 
902 

945 
990 

035 
08  1 

742 
782 
822 

864 
906 
950 

994 
040 
086 

746 
786 
827 

868 
911 
954 

999 
044 
091 

750 
790 
831 
872 
9i5 
959 
*oo3 
049 
095 

754 
794 
835 
877 

?i9 
03 

*oo8 
053 

100 

758 
798 

839 
881 
924 
967 

*OI2 
058 
105 

762 
802 
843 
885 
928 
972 
*oi7 
063 
no 

766 
806 
847 
889 
932 
976 

*O2I 
067 
114 

770 
8ro 
851 

894 
937 
981 

*026 

072 
119 

129 

133 

138 

143 

148 

153 

158 

162 

167 

172 
221 
272 

323 
376 
430 
485 

54i 
599 

177 
226 

277 

329 
38i 
435 
490 

547 
604 

182 
231 
282 

334 
387 
441 
496 

I52 
610 

187 
236 
287 

339 
392 
446 
502 
558 
616 

192 
241 
292 

344 
397 
452 

507 
564 
622 

197 
246 
297 
350 

403 
457 

513 
570 
628 

2O2 
252 
303 

355 
408 

463 
518 

I75 
634 

207 

257 
308 

360 
414 
468 

524 
581 

639 

211 
262 
3'3 
365 
419 

474 

% 

645 

216 
267 
3i8 

37i 
424 
479 

535 
593 
651 

657 
717 

779 
841 

905 
971 
0.03  037 

106 
175 

247 

663 

669 

675 

68  1 

687 

693 

699 

705 

711 

723 
785 
848 

912 

977 
044 

"3 
183 

254 

729 
791 
854 
918 
984 
051 

120 
100 
26l 

735 
797 
860 

925 
991 
058 

126 
197 
268 

742 
803 
867 

931 

22 

065 

133 
204 
276 

748 
810 

873 

938 
*oo4 
071 

140 

211 
283 

754 
816 
879 

944 
*on 
078 

147 
218 
290 

760 
822 
886 

95i 
"017 
085 

'54 

3 

766 
829 
892 

957 

*024 

092 

161 

232 
305 

772 
835 
899 

964 
*031 
099 

168 
240 
312 

320 

327 

334 

342 

349 

357 

364 

37i 

379 

386 

394 
470 

548 

627 
708 
790 

5* 

961 

0.04  049 

401 
478 
555 
635 
7i6 
799 
883 
970 
058 

409 
485 
563 

643 
724 
807 

892 

979 
067 

417 
493 
57i 
651 

732 
816 

901 

987 
076 

424 
501 

579 
659 

74i 
824 

909 
996 
085 

432 
509 

587 
667 

749 
832 

918 

*005 

094 

439 
516 

595 

675 
757 
841 

926 
*oi4 
103 

447 
524 
603 

683 
765 
849 

935 

*023 
112 

455 
532 
611 

691 

774 
858 

944 

*032 
121 

462 
540 
619 

700 
782 
866 

953 
*o4o 

130 

139 

148 

'57 

167 

176 

i*5 

194 

203 

213 

222 

A. 

B.   0 

2 

a 

4 

6 

tf 

7 

8 

9 

Prop.  Pts.   ! 

88 


TABLE  VI. 


ADD  f  Io8*  —  lo%a  =  A*          SUB  i  lo%a  "*  loS^  =  A 
x  i  log(«  +  J)  =  log*  +  ^.     bU  i  log(«  -  *)  =  log£  +  ^. 

A. 

B.  0 

1 

2 

a 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

0.00 

9.01 
9.02 

9.03 
9.04 
9.05 

9.06 

9.07 
9.08 
9.09 

9.10 

9.11 
9.12 
9-13 
9-H 

9.15 
9.16 

9-17 
9.18 
9.19 
9.20 

9.21 
9.22 
9-23 

9-24 
9.25 
9.26 

9-27 
9.28 
9.29 

9.30 

9.31 
9-32 
9-33 
9-34 
9-35 
9.36 

9-37 
9-38 
9-39 
9.40 

9.41 
9.42 
9-43 
9-44 
9-45 
9.46 

9-47 
9.48 
9-49 
9.50 

0.04  139 

148 

157 

167 

176 

185 

194 

203 

213 

306 
401 
499 
598 
700 
803 

909 
017 
127 

222 

z 

3 

3 
4 
5 
6 
7 
8 

9 

z 

3 

3 

4 
5 
6 

7 
8 

9 

i 

2 

3 
4 
5 
6 
7 
8 

9 

X 
2 

3 
4 
5 
6 

7 
8 

9 

z 

3 

3 
4 
5 
6 

7 
8 

9 

z 
2 
3 
4 
5 
6 

7 
8 

9 

9 

0.9 

1.8 
2.7 
3-6 
4-5 
5-4 
6-3 
7.2 
8.z 

xa 

Z.2 

2.4 
3.6 
4.8 
6.0 
7.2 
8.4 
9.6 
10.8 

'5 
»-5 

3-0 
4-5 
6.0 
7-5 
9.0 
10.5 

13.0 

»3-5 
18 
z.8 
3-6 
S-4 
7.2 
9.0 
10.8 

13.6 

H.4 

16.2 

at 

zo 
z.o 

2.0 

3.c 

4.0 
5.0 

6.0 
7.0 
8.0 
9.0 

'3 

1-3 

2.6 

3-9 

5-2 

6-5 
7.8 
9.1 
10.4 
zz.7 
16 
z.6 
3.2 
4.8 
6.4 
8.0 
9.6 

IX.  2 
13.8 

x4-4 

«9 

1.9 
3.8 
5-7 
7.6 
9-5 

II.  4 

T3-3 
15.2 
17.  x 

23 

XI 

z.z 

3.il 

3-3 
44 

5-5 
6.6 

7-7 
8.8 

9-9 
M 

*-4 

3.8 

4.  « 
5-6 
7.0 
8.4 
9.8 
ii.  a 

12.6 

17  ; 
»«f 

3-4 

5-1 

6.8 
8-5 
10.  a 
11.9 
13.6 
'5-3 
2O 
a.o 
4-0 
6.0 
8.0 

IO.O 
12.0 
14.0 

16.0- 
18.0 

as 

231 

325 
421 

5!9 
618 
720 

824 

93i 
0.05  039 

240 
334 
43° 
528 
628 
73i 

835 
941 
050 

250 

344 
440 

I38 
639 

74i 

845 
952 
061 

259 

353 
450 

548 
649 
75i 
856 

963 
072 

268 

363 
460 

I58 
659 

762 

867 
974 
083 

278 

373 
469 

568 
669 
772 

877 
985 

094 

287 
382 
479 

I78 
679 

782 
888 

995 
105 

297 
392 
489 

588 
689 
793 
898 
006 
116 

315 
411 

509 

608 
710 
814 

92O 
028 
139 

150 

161 

172 

183 

195 

206 

217 

229 

240 

251 

263 
378 
496 

616 

738 
863 

991 

0.06  121 

254 

274 
390 
508 

628 

$ 

004 

134 
267 

286 
401 
519 

640 
763 
889 

*oi7 

147 
281 

297 
4i3 
53i 
652 

775 
901 

*030 
161 

294 

308 
425 
543 

664 
788 
914 

1*043 

174 
308 

320 
436 
555 

677 
800 
927 
*o56 
187 
321 

332 
448 
567 

689 
813 
939 
*o69 
200 
335 

343 
460 

579 
701 
825 
952 

*082 

214 

348 

355 
472 

59i 

7H 
838 
965 

*095 
227 
362 

366 
484 
604 

726 

851 
978 

*io8 
240 
376 

389 

403 

417 

430 

444 

458 

472 

486 

500 

513 

13 

812 

959 
0.07  108 
261 

416 

575 
736 

54i 
683 
827 

973 
123 
276 

432 
59i 
753 

I55 
697 

841 
988 
138 
291 

448 
607 
769 

569 
711 
856 

*oc>3 
154 
307 

463 
623 
785 

583 
725 
870 

*oi8 
169 
322 

479 
639 
802 

597 
740 
885 

*033 
184 
338 

495 
655 
818 

612 

754 
900 

*048 
199 
354 

I11 
671 

835 

626 

769 
914 

*o63 
215 
369 

527 
687 
851 

640 

783 
929 

*o78 
230 

385 

543 
704 
868 

654 
798 
944 

*093 
245 
400 

559 
720 
884 

901 

918 

934 

95i 

968 

985 

*OOI 

*oi8 

*<>35 

*0$2 

0.08  069 
240 
415 
592 
774 
958 
0.09  146 
338 
533 

086 
257 
432 
610 
792 
977 
165 
357 
553 

I0j 

275 
450 
628 
810 
996 

184 

377 
573 

120 
292 
468 

646 
829 

*oi4 
204 
396 
593 

137 
309 
485 
664 

847 
*o33 

223 
416 
612 

154 
327 
503 
683 
865 

*052 

242 

I35 
632 

171 
344 
521 

701 
884 
*o7i 

261 

455 
652 

188 
362 
539 
719 

206 
379 

557 

737 
921 
*io8 

299 

494 
692 

223 

397 
574 

755 
940 

*I27 

319 

5H 
712 

902 
*090 

280 

474 
672 

4-2 

6.3 

8.4 

10.5 
12.6 

14.7 

16.8 
18.9 

24 

3.4 
4.8 
7.2 
9.6 

12.0 
14.4 

z6.8 
19.2 

21.6 

4-4 
6.6 
8.8 

II.  0 

13-2 
iS-4 

17.6 
19.8 

«5 

2-5 
5.0 
7-5 

10.  0 

12.5 
15.0 
17-5 

20.0 
22.5 

4-6 
6.9 
9.3 
XZ.J 

13.8 

i6.z 

18.4 
20.7 

26 

2.6 

5-2 

7.8 

10.4 
13-0 

15.6 

lS.2 
20.8 

23-4 

732 

935 
o.io  141 

35i 

565 
783 
o.n  005 

23 

46 
69- 

933 

752 

773 

793 

813 

833 

853 

874 

894 

914 

955 
162 

373 

587 
805 
028 

% 

715 

976 
183 
394 
609 
827 
050 

277 
507 
742 

996 
204 
415 
630 
849 
073 
3oc 

53i 
766 

*oi7 
225 
437 
652 
872 
095 

323 

554 
790 

*038 
246 
458 

674 
894 
118 

345 

*os8 
267 
479 
696 
916 
140 

368 
60  1 
837 

*079 

288 
501 

718 
938 
163 

392 

62; 

86 

*IOO 

309 
522 

739 
960 
1  86 

415 
648 
885 

*I2O 
330 

544 
761 

983 
208 

438 
671 
909 

957 

98 

*oc»5 

*030 

*054 

*o78 

*I02 

*I27 

*i5i 

A. 

B.  0 

1 

2 

8 

| 

6 

6 

7 

8 

9 

Prop.  Pts. 

ADDITION  AND  SUBTRACTION  LOGARITHMS. 


89 


A   (  log£  —  loga  =  A.          9   i  loga  —  log£  =  B.     \ 
x  \  log  (a  +  t)  =  loga  +  B.      bUB*  i  log(a  -  b]  =  log^  +  A 

A. 

B.   0 

1 

2 

a 

4 

5 

6 

7 

*I02 

8 

9 

Prop.  Pts. 

9.50 

9.51 
9.52 
9-53 
9-54 
9-55 
9.56 

9-57 
9.58 
9-59 
9.60 

9.61 
9.62 
9-63 
9.64 
9.65 
9.66 

9.67 
9.68 
9.69 

9.70 

9.71 
9-72 
9-73 
9-74 
9-75 
9.76 

9-77 
9.78 
9-79 
9.80 

9.81 
9.82 
9-83 
9.84 
9.85 
9.86 

9-8'/ 
9.88 
9.89 

9.90 

9.91 
9.92 
9-93 

9-94 
9-95 
9.96 

9-97 
9.98 

9-99 
0.00 

o."  933 
0.12  175 
422 
673 
928 
0.13  188 
452 
721 

994 
0.14  272 

~554 
841 
0.15  133 
43° 

,  73i 
o.io  037 

349 
665 
986 
o.i7_3i2 

643 
980 
0.18  322 
668 

0.19  020 

378 
740 

0.20  108 

481 
860 

957 

981 

*oo5 

*O3O 

*°54 

*078 

*I27 

"151 

i 

2 

3 
4 
5 
6 
7 
8 

9 

i 

2 

3 
4 
5 
6 

7 

8 

9 

i 

2 

3 
4 
5 
6 
7 
8 

9 

i 

2 

3 
4 

5 
6 
7 
8 
9 

i 

5 

3 
4 

5 

e 

7 
8 

9 

i 

2 

3 
4 

6 

7 
8 
9 

37 

2.7 

5-4 
8.1 
10.8 

13-5 
16.2 
18.9 

21.6 

24-3 
31 

3-i 

6.2 

9-3 
12.4 

15-5 
18.6 
21.7 
24.8 
27.9 

35 

3-5 
7.0 
10.5 
14.0 
17-5 

21.0 
24-5 
28.0 
31-5 

39 

3-9 
7.8 
11.7 
15-6 
19-5 
23-4 
27-3 
31-2 
35-  1 
43 
4-3 
8.6 
12.9 
17.2 
21.5 
25.8 
30-1 
34-4 
33.7 
47 
4-7 
9-4 
14.1 
18.8 
23-5 
28.2 
32.9 
37-6 
43.3 

38 
2.8 

5-6 
8.4 

II.  2 
I4.0 

16.8 
19.6 
22.4 
25.2 

32 

3-2 

6.4 
9-6 

12.8 

16.0 
19.2 
22.4 
25.6 
28.8 

36 

3-6 

7-2 

10.8 
14.4 
18.0 

21.6 

25.2 
28.8 
32-4 
40 

29 

2-9 

5-8 
8.7 
ix.  6 
14-5 
17.4 
20.3 
23.2 
26.1 

33 

3-3 
6.6 

9-9 
13.2 
16.5 
19.8 
23.1 
26.4 
29.7 

37 

3-7 
7-4 
u.  i 
14.8 
18.5 

22.2 

25-9 
29.6 

33-3 
41 

30 

3-o 
6.0 

9  a 

12  0  ' 

15  o 
18.0 

21.0 
24.0 
27.0 

34 
3-4 
6.8 

10.2 
I3.6 
17.0 
20.4 
23.8 
27.a 
30.6 

38 
3-8 

7.6 
11.4 
15.2 
19.0 

22.3 
26.6 
30.4 

34-a 

43 

200 

447 
698 

954 

214 
479 
748 

*O2I 
300 

224 
472 
724 
980 
240 
505 

775 
*049 
328 

249 
497 
749 
*oo6 
267 
532 
802 
*077 
356 

274 
522 
775 

*032 
293 

559 
829 
*I04 
384 

298 

547 
800 

*o58 

3J9 
586 

85* 

*I32 

412 

323 
572 
826 

*o84 
346 
613 
884 
*i6o 
441 

348 

597 
851 

*IIO 

372 
640 

911 

*i88 
469 

372 
622 
877 
*I36 

399 
667 

939 

*2l6 

497 

397 
648 

903 

*l62 

425 
694 

966 

*244 
526 

583 

611 

640 

668 

697 

726 

755 

783 

812 

870 
162 

460 

76l 
068 
380 

697 

*oi8 
345 

899 
192 

489 

792 
099 
411 

729 
*o5i 
378 

928 

221 
520 

822 
I30 

443 
761 
*o83 
411 

957 
251 

550 

8I3 
161 

474 

793 
*ii6 

444 

986 
281 
580 

884 
192 
506 

825 
*I48 
477 

*oi6 
310 
610 

914 

224 
538 

857 
*i8i 
510 

*045 
340 
640 

945 

255 
569 

889 

*2I4 

544 

*074 
370 
670 

976 
286 
601 

921 

*247 
577 

*IO4 
400 
701 

*oo7 
317 
633 

954 
*279 
610 

677 

710 

744 

777 

811 
*i5o 

494 
844 

198 
558 
923 
294 
670 
^052 

845 
*i84 
529 
879 

234 

¥ 
96o 

331 

708 

*O90 

878 

912 

946 

*oi4 
356 
703 
056 
414 
777 

145 
519 
898 

*048 
390 
738 
091 
450 
813 
182 
557 
937 

*082 

425 
773 
127 
486 
850 

220 

594 
975 

*ii6 
460 
808 

163 
522 
887 

257 

*632 
*oi3 

*2l8 

564 
914 

270 
631 

997 

369 
746 

*I28 

*253 
599 
949 
306 
667 
*Q34 

406 
784 
*i67 

*287 
633 
985 

342 
704 
*o7i 

444 
822 

*206 

8.0 

12.0 

16.0 
20.  o 

24.0 
28.0 
32.0 

36.0 

44 

4-4 
8.8 
13-2 
17.6 

22.0 
26.4 
30.8 

35-2 
39.6 

48 

4.8 
9.6 
14.4 
19.2 
24.0 
28.8 
33-6 
38.4 
43-2 

8.2 

12.3 
16.4 
20.5 
24-6 
28.7 
32.8 
36-9 
45 
4-5 
9.0 
13-5 
18.0 
22.5 
27.0 
3i-5 
36.0 

40-5 

49 

4.9 
9.8 
14.7 
19.6 
24-5 
29.4 
34-3 
39-2 
44.1 

8.4 

12.6 

16.8 

21.0 

25.2 
29.4 
33-6 

37-8 
46 
4.6 
9.2 
13-8 
18.4 
23.0 
27.6 
32.2 
36.8 
41.4 

50 
5-0 

IO.O 

iS-o. 
20.  o 

25.0 
30.0 

35-0 
40.0 
45-0 

0.21  244 

634 
O.22  O29 

430 
836 
0.23  247 
665 

0.24  088 
5I6 
950 

283 

322 

361 

399 

438 

477 

516 

556 

595 
989 
3S9 
795 

*206 

623 
*Q45 

473 
907 

*346 

673 
069 
470 

877 
289 
707 

130 
559 
994 

712 
109 
510 

918 
330 
749 

173 
603 
*o38 

752 
149 

55i 

959 
372 
791 

216 
646 

*082 

791 
189 
59i 

*000 

414 

833 

258 
689 

*I26 

831 
229 
632 

*04I 
455 
875 
301 
733 

*I70 

870 
269 
673 

*082 

497 
918 

344 
776 

*2I4 

910 
309 
713 

*I23 

539 
960 

387 
819 
*258 

949 
349 
754 
*i65 
581 
*oo3 

430 
863 

*302 

0.25  390 

434 

479 

523 

568 

612 

657 

701 

746 

791 

836 
0.26  287 

744 
0.27  207 

675 
0.28  149 

629 
0.29  115 
606 

881 
332 
790 

253 
722 

197 

677 
163 

655 

926 
378 
836 

300 
769 
245 

726 

212 
705 

970 

423 
882 

346 
817 
292 

724 
261 

754 

*oi6 
469 
928 

393 
864 

340 
822 
310 

804 

*o6i 
515 
974 
440 
911 
388 

871 
359 
854 

*io6 
560 

*02I 
487 

959 
436 
920 
409 
903 

*i5i 
606 
"067 

534 
*oo6 
484 

968 
458 
953 

*i96 
652 
*U4 
581 
*Q54 
532 
*oi7 
507 
*oo3 

*242 
698 

*i6o 
628 

*IOI 

581 

*o66 
556 
*053 

0.30  103 

153 

2O3 

253 

303 

354 

404 

454 

505 

555 

1  A. 

B.   0 

1 

2 

a 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

0 

ADD  {  ^ga-logd  =  A.          g   (  loga  -  log£  =  B. 
'  \  log  (a  +  &)  =  log  £  +  -#.        |  log  (a  —  £)  =  log  £  +  A. 

A. 

B.  0 

1 

2 

3 

4 

6 

0 

7 

8 

9 

Prop.  Pts. 

0.00 

O.OI 
O.O2 
0.03 

0.04 
O.O5 
0.06 

0.07 
0.08 
O.O9 

0.10 

O.II 
O.I2 

0.13 

0.14 
0.15 
0.16 

0.17 
0.18 
0.19 

0.20 

0.21 
0.22 
0.23 

0.24 
O.2J 

O.26 

0.27 
0.29 

0.30 

0.31 
0.32 
0.33 

0-34 
0.35 
0.36 

0.37 
0.38 
0.39 

0.40 

0.4 

0.42 
0.43 

0.4 

0.4 
0.46 

0.4 
0.4 
0.4 

0.5 

0.30  103 
606 
0.31  115 
629 

0.32  149 

675 
0.33  207 

744 
0.34  287 
836 

153 

203 

253 

303 

354 

404 

454 

505 

555 

5° 

5-0 

10.0 

15.0 

20.0 
25.0 
30.0 

35-o 
40.0 
>  5-o 
54 
5-4 

!  10.8 
JI6.2 
J2I.6 

527.0 
532.4 
737-8 
3  43-2 
?48.6 

58 
'  5-8 

2  II.  6 

3  17-4 
4  23.2 
5  29.0 
634.8 
740.6 
846.4 
952.2 

62 

I   6.2 
2  12.4 

3  18.6 
424.8 
53i-o 
637.2 
743-4 
849-6 
955.8 

66 

i  6.6 
213.2 

3  19-8 
426.4 
533-c 
639-6 
746.2 
852.8 
959-4 
70 

2  14.  c 

32I.C 

4  28.  c 
535-c 
6  42.  c 

749  < 
856.c 

963.* 

51 

5-i 

tO.  2 

'5-3 
20.4 

25-5 
30.6 

J5-7 
to.  8 
15-9 
55 

5-5 

II.  0 

'6.5 

22.0 
27.5 

33-o 
38.5 
44-o 
49-5 
59 

5-9 
ii.  8 
17.7 
23-6 
29-5 
35-4 
4i-3 
47-2 
S3-i 
63 
6-3 

12.6 

18.9 

25.2 

37-8 
44.1 
5°-4 
56-7 
67 
6-7 
13-4 

20.1 
26.8 

33-5 
40.2 
46-9 
53-6 
60.3 

71 

14.2 
21.3 
28.4 

35.5 

42.  e 

49-3 
56.8 

63-<3 

5-2 

to.  4 

20.8 

26.0 
;i.2 
56.4 

56 

5-6 

[1.2 

[6.8 
22.4 
28.0 
33-6 
39-2 
44-8 
50.4 

60 
6.0 

12.0 

18.0 

24.0 
30.0 
36.0 

42.0 

48.0 

54-0 
6.4 

12.8 

19.2 
25.6 

32.0 

38.4 

44.8 
51.2 
57-6 
68 
6.8 
13.6 
20.4 
27.2 
34-c 
40.8 
47-C 
54-4 
61.2 

73 

7.2 
14.4 

21.  e 

28.  £ 

43-s 
50.4 
57-< 
64.* 

53 

5-3 
0.6 

5-9 

1.2 

6-5 
1.8 
7-J 

2-4 

7-7 
57 
5-7 
M 
7-i 

22.8 

28.5 

34-a 
39-9 
45-6 
Si-3 
6x 
6.x 

12.2 
I8.3 
24.4 
30-5 
36.6 
42.7 
48.8 

54-9 
65 
6-5 
13-0 
19.5 
26.0 
32-5 
39-0 
45.5 
52.0 
58.5 
69 
6.9 
13-8 
20.7 
27.6 
34-5 
41.4 
48.3 

55-2 
62.1 

73 

7-3 
14.6 
21.9 
29.9 
36.5 
43-8 
51.1 
58.4 
65-7 

656 
1  66 
68  1 

20  1 
728 
260 

798 
342 
891 

707 
217 

732 

254 

852 

396 
946 

758 
268 

784 
306 
834 
367 
906 

OOI 

809 
320 
836 

359 
887 
421 

960 
*5°6 

859 
37i 
888 

411 
940 

474 

°'5 
561 

112 

910 
422 
940 

464 

993 
528 

069 
616 
168 

961 
474 
992 

5I2 

046 
582 

123 
670 
223 

012 
526 
045 

569 
100 

636 

178 
726 

279 

063 

577 
097 

622 

153 
690 

232 
78i 
334 

0.35  390 

446 

502 

558 

614 

670 

726 

782 

838 

894 

*95? 
0.36  516 

0.37  088 
665 
0.38  247 
836 

0.39  430 
0.40  029 

634 
0.41  244 

007 
573 

723 
306 
895 

489 
089 
695 

*o63 
630 
203 

363 
954 

549 
149 
756 

119 

687 
260 

839 
423 
*oi3 

609 

2IO 

816 
"428 

669 
297 

570 
214 
864 

•?J! 

744 

897 
482 

*073 
669 
270 
877 

*233 
80  1 
375 
955 

*I32 

729 

33o 

938 

+289 
858 
433 
*oi4 
600 
*I9I 

789 
999 

*346 
916 
491 

*072 

/59 
849 

452 

*o6i 

403 
973 
549 

718 
*3io 

909 
512 

*I22 

*459 
*O3O 
607 

777 
*370 

969 

573 
*i83 

306 

367 

490 

552 

613 

1 

487 

*I22 
763 
408 

*o6o 
716 
"377 

675 

737 

798 

860 
0.42  481 
0.43  108 
740 
o.44  378 
0.45  020 

668 
0.46  322 

980 

922 

544 
171 

804 

442 
085 

387 

£O/1  ( 

984 
606 

234 

867 
506 
149 

799 

453 

*II2 

*io8 

360 

995 
634 
279 

929 
584 

*245 

*i70 
794 
423 
*o58 
698 
344 

994 
650 
*3n 

920 
550 
*i86 
827 

473 

782 
*444 

*357 
982 
613 

89? 
538 

*IGO 
848 
*5IO 

*045 
677 

*3H 
956 
603 

*256 

914 

*577 

Q.47  643 
0.48  312 
986 
0.49  665 

0.50  349 
0.51  037 

0.52  430 
o.53  133 
84 

o.54  554 

710 

777 

844 

910 

977 

O4^ 

*iii 

*I78 

*245 
918 
*597 

*280 

968 
661 
*36o 

*062 

770 
*483 

*379 

73: 

417 
107 

801 

500 
204 
912 

447 

*I2I 

Soi 

486 
176 
870 

570 
274 
983 

869 

555 
245 
940 

640 
345 
*°55 

*257 
938 
624 

*OIO 

710 

416 

*I26 

*64- 
*oo6 
692 

384 
*o8o 

78 
486 
*i97 

716 
*393 
*o74 
761 

453 
"150 

85 

*M78 

783 
*46i 

*I43 
830 
522 

*220 

92 
628 

*34o 

851 
*529 

*2II 

899 

592 

*289 
992 
699 
*4i 

626 

697 

769 

841 

912 

984 

*os6 

*I28 

*2OO 

0.55  272 

,  994 
0.56  72 

o.57  45 
0.58  18 
92 

0.59  67 
0.60  42 
0.61  17 

344 
*o66 

794 
262 

748 
497 
251 

416 
*i39 
86 

59 

3 

82 

57 
32 

488 

*2II 

940 

672 
4ic 
*i5i 

897 
648 
402 

*28° 

746 

484 

*226 
972 

723 
478 

632 
*357 
*o86 

819 

558 
*3oo 

79 

554 

704 

*?59 

893 
63 
*37~ 

*I2 

874 

63c 

777 

*502 
*232 

967 
706 

*449 
*i9 

94 
70 

*46 

849 

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*3o 

*040 

*78C 

*27 

784 

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*379 
*ii4 
854 
*S98 

*347 

*IOO 

857 

*ooc 

*o8 

*i6 

*237 

*3M 

*39c 

*54 

*6i9 

B.  0 

1 

2 

3 

4 

5 

0 

1 

8 

9 

Prop.  Pts. 

ADDITION  AND  SUBTRACTION  LOGARITHMS. 


A   f  loga  —  log  b=  A.          q   (  loga  —  log£  =  B. 
ADD'  i  log(a  +  b)  =  log*  +  B.       3'  i  log(0  -  b)  =  log£  +  ^. 

A. 

B.  0 

1 

2 

a 

4 

5 

G 

7 

8 

9 

Prop.  Pts. 

0.50 

0.51 
0.52 
o.53 
0.54 
o.55 
0.56 

0.57 
0.58 
0.59 

0.60 

0.61 
0.62 
0.63 

0.64 
0.65 
0.66 

0.67 
0.68 
0.69 

0.70 

0.71 
0.72 
o.73 
0.74 
0.75 
0.76 

0.77 
0.78 

o.79 
0.80 

0.81 
10.82 
0.83 

0.84 
0.85 
0.86 

0.87 
0.88 
0.89 

0.90 

0.91 
0.92 
0.93 

0.94 
0.95 
0.96 

0.97 
0.98 
0.99 

1.00 

0.61  933 

*cx>9 

+085 

*l6i 

*237 

*3i4 

*39o 

*466 

*542 

*6i9 

X 
2 

3 
4 
5 
6 

7 

8 

9 

X 
3 

3 
4 
5 
6 

7 

8 

9 

t 
a 
3 
4 
5 
6 
7 
8 
9 

X 
2 
3 
4 
5 
6 
7 
8 

9 

X 
3 

3 
4 
5 
6 

7 
8 

9 

X 

a 
3 
4 
5 
6 
7 
8 
o 

74 

7-4 
14-8 

22.2 
29.6 
37.0 

44-4 
51.8 

59-2 
66.6 

77 
7-7 
15-4 

30.8 
38.5 
46.2 

53-9 
61.6 
69.3 
80 
8.0 
16.0 
24.0 
32.0 
40.0 
48.0 
56.0 
64.0 
72.0 

83 

8-3 
16.6 
24.9 
33-2 
41.5 
49-8 
58.1 
66.4 
74-7 
86 
8.6 
17.2 
25.8 
34-4 

51.6 
60.2 
68.8 
77-4 

89 
89 

2^7 
35.  6 
44-5 
53-4 
62.3 
7».a 
80.1 

75 

7-5 
15-0 
22.5 
30.0 
37-5 
45-0 
52-5 
60.0 
67-5 
78 
7.8 
15.6 
23-4 
31.2 

39-0 
46.8 
54-6 
63.4 
70.2 

81 
8.x 
16.2 
24.3 
32.4 
40.5 
48.6 
56.7 
64.8 
72.9 

84 

8.4 

16.8 
25.2 

33-6 
42.0 

50.4 
58.8 
67.2 
75-6 
87 
8-7 
17-4 
26.1 
34-8 
43-5 
52.2 
60.9 
69.6 
78.3 
90 
9.0 
18.0 
27.0 
36.0 
45  .e 
54-c 
63.0 
72.0 
8x.c 

76 
7.6 
15-2 

22.8     I 
30-4 
38.0     | 

45-6 
53-2 
60.8 

68.4 

79 
7-9 
15.8 
23.7 
31.6 
39-5 
47-4 
55-3 
63.2 
71.1 
83 
8.2 

16.4 
24.6 
32.8 

41.0 
49.2 

57-4   ! 
65.6 
73-8 

85 

8.5 
17.0 
25.5 
34-o 
42.5 
51-0 
59-5 
68.0 
76.5 
88 
8.8 
17.6 
26.4 

35-a 
44.0 
52.8 
61.6 
70.4 

9* 
9.1 
18.2 
27.3 
36.4 
45-5 
54-6 
63.7 
72.8 
81.9 

0.62  695 
0.63  461 
0.64  231 

0.65  005 

783 
0.66  565 

0.67  351 
0.68  141 
935 

771 
538 
308 

083 
861 
644 

430 

220 

*oi4 

848 
615 
386 

160 

939 
722 

509 
*3°° 

924 
692 
463 

238 
*oi8 
Soi 

588 
379 
*I74 

*OOI 

768 
540 

"096 
879 
667 
458 

845 
618 

394 
*i74 

958 

746 
538 
*333 

*I54 
923 
695 
472 

*252 

617 
*4I3 

*23I 

*ooo 
773 

549 
*33° 
*ii5 

904 
696 
*493 

*3°7 
"077 
850 

*627 
*I94 

983 
776 

*573 

OOOO  W4^vjvO-iU> 
ononONVJOOO  tOOnOO 
to  on  U  U>  ^>J  on  G04-  4- 

0.69  732 

812 

892 

972 

*0$2 

*I32 

*2I2 

*293 

*373 

*453 

0.70  533 
0.71  338 
0.72  146 

958 
0.73  774 
o.74  592 

o.75  415 
0.76  240 
0.77  069 

_9oi 

0.78  736 

o.79  575 
0.80  416 

0.81  261 
0.82  108 
959 
0.83  812 
0.84  668 
0.85  527 

614 
419 
227 

855 

674 

497 
323 
152 

694 

499 
308 

*I2I 

937 
757 

579 
406 

235 

774 
580 

390 

*202 

839 
662 
488 
318 

*IOI 

921 

744 
57i 
401 

935 

742 
552 
*365 
*i83 

*003 

827 
654 
485 

*oi6 

823 

633 

*447 

*o8s 
909 

737 
568 

*o96 
904 
7H 

*346 
*i68 

992 
820 
651 

*I77 
984 
796 

*6io 
*428 

*o75 
903 
734 

*257 

877 
^692 

-"157 
818 

984 

820 
659 
500 

345 
193 
*044 
898 

754 
613 

*o68 

*I5I 

*235 

*3i8 

*402 

*48$ 

*569 

*653 

904 

743 
585 

430 

278 

*I29 

983 
840 
700 

987 
827 
669 

515 
363 
*2I4 

926 
786 

911 

754 

599 
448 
*3oo 

*I54 

*OI2 
872 

995 
838 

684 
*533 

*240 
958 

*239 

922 
769 

618 
*47o 

*i63 
*oo7 

854 
703 
*556 

*4H 

938 

788 

*497 
*355 

*2I7 

*49i 
*?76 

*023 

873 

*727 

*44i 

0.86  389 

476 

562 

648 

735 

821 

908 

994 

*o8i 

"167 

0.87  254 
0.88  121 
991 

0.89  863 
0.90  738 
0.91  616 

0.92  496 
0.93  378 
0.94  263 

340 

20£ 

951 
826 
704 

584 
466 

427 
295 

914 

79I 

672 

555 
440 

382 

*252 

*I25 
*OOI 

879 
760 
643 

529 

600 
469 
*339 

*2I3 

967 
848 
732 
617 

687 
*556 

!3°° 
*I77 

*°55 

936 
820 
706 

774 

*643 
*5H 

*'4: 

£08 
795 

861 

730 
*6oi 

*475 
*352 

*23I 

*ii3 
997 
883 

947 

*687 
*563 
t440 

*20I 

*o86 
972 

904 
*776 

*4o8 

*I74 
*o6i 

950 

Q.95  '50 
0.96  039 

0.97  82^ 

0.98  720 
0.99  618 
i.  oo  519 

i.oi  421 

1.02  325 
1.03  231 

239 
128 

*020 
914 

810 
708 
609 

5" 
322 

327 

416 

505 

594 

683 

772 

861 

217 
*i09 

*003 

900 

798 
699 

601 
506 
413 

*i°8 

789 
692 
597 
503 

395 
*288 

*l82 

879 
782 

687 

594 

485 
*377 

*272 

*i69 
*o68 
969 

873 
778 
685 

*362 

*259 
*I58 
*o6o 

963 
868 
776 

663 

*45i 
*349 

959 
867 

*645 
*54i 

*44 
957 

841 

*735 
*63i 

*528 
*428 

"•140 

1.04  139 

230 

321 

412 

503 

594 

685 

776 

867 

958 

1  A. 

B.   0 

1 

2 

8 

4 

6 

G 

7 

8 

9 

Prop.  Pts. 

2                          TABLE  VI. 

ADD  )  l°Za  ~~  10g  b  =  A'               QTTR  f  log  *  ~  lo§  *  =  •#• 

*  t  log  (a  +  b)  ==  log  ^+  A         i  log(«  -  J)  =  log^  4-  A. 

;  A. 

B.   0 

1 

2 

a 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

1.00 

01 

02 

03 

04 

°l 

06 

:>7 
08 
09 

10 

ii 

12 
13 

14 
15 

16 

17 
18 

19 
.20 

21 
22 
.23 

.24 
.25 
.26 

!28 
.29 

.30 

•  31 
•  32 
<  -33 

:   -34 
i   -35 
•36 

•37 
•38 
•  39 

1.04  139 

230 

321 

412 

503 

594 

685 

776 

867 

958 

9 

i  9 

2  18 
3  27 
4  36 
5  45 
6  54 

9  Si 
I 

2 

3 
4 

i 
I 

9 
i 

2 

3 
4 

1 

9 

! 

i  c 

2  I< 

32i 

4  3^ 
5  4 

6  \ 
7  6( 

8  7 
9  8 

i 

2 

3 
4 

.1 
I 

9 

i 

.2 

•3 
•4 

•9 

93 

27 

1 

65 

g 

9 

,i 

28 

37 

1 
1 

)5 

)-5 
).o 

1:1 

J-S 
7-c 

It 

5-5 

? 
5 
IS 

2C 

1 
6; 

7' 

8; 

9* 

9.2 
18.4 
27.6 
36.8 
46.0 

55-2 
64.4 

g;l 
I 

q 
2 

I 

I 

4 
7 

\ 

i 

.2 

.6 

.0 

•i 

.2 

.6 

96 
96 
19.2 
28.8 

38.4 
48  o 
57-6 
67.2 
76.8 
86.4 

>7 

>-7 
>-4 

!i 

ii 

',i 

r-3 

1.05  049 
961 
i.  06  875 

1.07  790 
i.  08  708 
1.09  627 

i.  10  548 
i.  i  i  470 
1.  12  394 

140 

053 
966 

882 
800 
719 

640 
562 
486 

232 
144 

058 

974 
891 
811 

732 
655 
579 

323 
235 
149 

065 
983 
903 
824 

747 
671 

414 
*326 

*24I 
I157 

*075 
995 
916 
839 
764 

505 
418 

332 

249 
167 
087 

009 
932 
857 

596 
*5o9 

*424 

34i 
259 
179 

101 

024 
949 

687 
"601 
*5i6 

*432 
*35i 

*27  1 

^193 
*ii7 

^042 

779 
692 
607 

524 
443 
363 
285 
209 
134 

870 

783 
699 

616 
535 
455 
378 
301 
227 

1.13  320 

412 

503 

598 

690 

783 

876 

968 

06  1 

i54 

i  •  14  247 
I.I5  175 
1.16  106 

1.17  °J7 
97i 
1.18  905 

1.19  841 
1.20  779 

1.  21  717 
1.22  657 

1.23  599 
1.24  54i 
1.25  485 

1.26  430 
1.27  376 
1.28  323 

1.29  272 

1.30  221 

1.31  J72 

340 
268 
199 

131 

064 

999 

935 
872 
811 

432 
361 
292 

224 

157 
092 

029 
966 
905 

525 
454 
385 

317 
251 
1  86 

122 

060 

999 

618 
547 
473 

411 

*344 
*279 

*2l6 

*I54 

*093 

I11 

640 

57i 
504 
*438 
*373 
*3ip 
*248 
*i87 

804 

733 
665 

597 

*S?! 

*467 

*403 
*342 

*28l 

897 
826 
758 
691 

I6? 
*56o 

*497 
*435 
*375 

990 

92O 
85I 

784 
718 
654 

591 
529 

469 

083 
013 
944 

877 
812 

748 

685 
623 
563 

751 

845 

939 

*Q34 

*I28 

*222 

*3i6 

410 

504 

693 
635 
579 
524 

47i 
418 

367 
316 
267 

787 
730 
674 

619 
565 
5J3 
462 
411 
362 

88  1 
824 

768 

714 
660 
608 

557 
507 
458 

975 
918 
863 

808 
755 
703 
652 
602 
553 

*070 
*oi3 
957 

903 
850 

797 
746 

697 
648 

*l64 

*I07 

*052 

997 
944 
892 

841 
792 

743 

*258 
*2O2 
*I46 
*092 

*d^9 
987 
936 
887 
838 

352 

296 
*24I 

*i87 
*i34 

*082 

•$ 

933 

447 
390 
*335 

*28l 
*229 

*I77 

*I26 

*077 

*029 

1.32  124 

219 

3H 

410 

505 

600 

695 

791 

886 

98l 

1.33  °77 
1.34  °3° 

985 

1.35  94i 
1.36  898 
1.37  856 
1.38  814 
1.39  774 
1.40  734 

172 
126 
*o8i 

*Q37 
994 
95i 
910 
870 
830 

267 

221 

*i76 

*I32 

*o89 
*O47 

*oo6 
966 
926 

363 
3i7 

*272 
*228 

*i85 
*i43 

*IO2 
*062 
*022 

458 

*4^2 
*367 

*324 
*28l 

*239 

*i98 
*I58 
*u9 

55^ 
508 

*463 
*4i9 
*377 
*335 

*294 

*254 

*2I5 

649 
603 

*559 

*5i5 
*472 

*43I 

*39o 
*35o 
*3ii 

744 
699 
*654 

*6n 

*568 

*527 
^486 
*446 
*4o7 

840 
794 
*750 

*7o6 
*664 
622 

*582 

*542 
*5o3 

935 

890 

*845 

*802 

*76o 
*7i8 

*678 
*638 
*599 

.40 

•4 
.42 

•44 
.4 
.46 

.47 
•  4 
.4 
1.50 

i  41  695 

792 

888 

984 

*oSo 

*I76 

*273 

*369 

*465 
*428 

*39 
*356 

*32 
*287 
*254 

*22 

*i8 
*i5 

*56i 

1.42  658 
1.43  62J 
1.44  584 

1.45  549 
I.465H 
1.47  480 

1.48  447 
1.49  4i 
1.50  383 

754 
717 
681 

645 
61 

577 
544 

% 

850 
813 
777 

742 
707 
674 

64 

608 
577 

946 
9IC 
874 
838 
804 

770 

737 

705 

674 

*043 
*oo6 
970 

935 
901 
867 

834 
802 

771 

*I39 

*I02 

*o66 

*o3 
997 
964 

93 

895 
868 

*235 
*I99 
*i63 

*I2S 

*o94 
*o6c 

*028 

996 
964 

*332 

*29 

*259 

*22~ 
*I9C 

*i5 

*I24 

*09 
*o6 

*524 
*488 
*452 
*4i8 
*384 
*350 

*3i8 
*286 

*255 

1.51  35 

449 

546 

64: 

740 

837 

934 

*o3 

*I2 

*225 

A. 

B.  0 

1 

2 

3 

4 

5 

0 

1 

8 

t) 

Prop. 

Pts. 

ADDITION  AND  SUBTRACTION  LOGARITHMS. 


93 


A   (  log  a  —  -  log  b  =  A.         Q   (  log  a  —  log  b  =  B. 
x  \  log  (a  +  J)  =  log<*  +  ^.     bu  B'  i  log(«  -  £)  =  log*  +  A. 

A. 
1.50 

•51 

•  52 
•  53 

•54 

:ii 

? 

•59 
.GO 

.61 

.62 
63 

.64 
•  65 
.66 

.67 
.68 
.69 

.70 

•  71 
.72 
•  73 
•  74 

:3 

•  77 
.78 
•79 
.80 

.81 
.82 
.83 

.84 
.85 
.86 

.87 
.88 
.89 

1.90 

•  91 
.92 

•93 

•94 
•95 
.96 

•97 
.98 
•99 
2.00 

B.  0 

1 

2 

546 

3 

643 

4 

5 

0 

7 

8 

9 

Prop.  Pts. 

1-51  352 
1.52  322 
1.53  292 
i  .  54  263 

1-55  235 
1.56  207 
1.57  180 

1-58  153 
1.59  128 

I.  60  102 

1.61  077 

449 

740 

837 

934 

*03I 

*I28 

*225 

i 

2 

3 

1 

I 
9 

i 

2 

3 
4 

i 
I 

9 

i 

2 

3 

4 

1 

9 

97 

9-7 
19.4 

38.8 

485 
58.2 
67.0 
77.6 
87.3 

98 

9-8 
19.6 
29.4 

39-2 
49.0 

68^6 

78.4 
88.2 

99 

9>i 

19.8 
29.7 
39  6 
49.5 

§1 
g;! 

419 

389 
360 

332 
304 
277 
251 
225 
200 

516 
486 
457 

429 
402 

375 

348 
322 
297 

613 

583 
555 
526 

499 
472 

446 
420 

395 

710 
680 
652 

624 
596 
569 

543 
517 
492 

807 
778 
749 
721 
693 
667 

640 

6i5 
590 

904 

875 
846 

818 
791 
764 

738 
712 
687 

*OOI 

972 

943 

861 

835 
810 

785 

*o98 
*o69 
*O4O 

*OI3 

985 
959 

933 
907 
882 

::n 

*I38 

*IIO 

*o83 
^056 

*O3O 

^005 
98o 

175 

273 

370 

468 

565 

663 

760 

858 

956 

1.62  053 
1.63  o3o 
i  64  006 

984 
1.05  962 
i  .  66  940 

1.67  919 
1.68  898 
1.69  878 

151 
127 
104 

*o8i 
*059 
*o38 

*oi7 
996 
976 

248 
225 

202 

*I79 
*I57 
*I36 

*ii5 
*094 
*074 

346 
322 
299 

*277 
*255 
*233 

*2I2 
*I92 
*I72 

444 
420 

397 

*375 
*353 
*33' 

*3IO 
*290 
*270 

54i 
518 

495 

*473 
*45i 
*429 

*4o8 
*388 
*368 

*348 

639 
616 

593 

*57o 
*548 
*527 

*5o6 
*486 
*466 

737 
713 
69o 

*668 
*646 
*625 

*6o4 
*584 
*564 

834 
811 
788 

*766 
*744 
*723 

*702 

*682 
*662 

932 

886 

*864 
*842 

*82I 

*8oo 
*78o 
*76o 

1.70  858 

956 

*054 

*I52 

*250 

*446 

*544 

*642 

*74i 

1.71  859 
1.72  820 
1.73  801 

1-74  783 
1.75  766 
1.76  748 

1-77  73i 
1.78  715 
1-79  699 

937 
918 
899 

881 
864 
847 
83o 
813 
797 

*o35 
*oi6 
998 
980 
962 
945 
928 

f£ 

*i33 
*ii4 
*o96 

*078 
*o6o 
*Q43 

*026 
*OIO 

994 

*23I 
*2I2 

*I94 

*I76 
*I59 
*i4i 

*I25 

*io8 

*092 

*329 

*3IO 
*292 

*274 
*257 
*24O 

*223 
*2O7 

*i9i 

*427 
*4©9 
*39o 

*373 
*355 
*338 

*32I 

*3os 

*289 

*525 

*507 
*489 
*47i 

*453 
*436 

*420 

*4o3 
*388 

*623 

*6o5 
*587 

*569 
*552 
*535 
*5i8 

*502 

*486 

*722 

*703 
*685 

*667 
*65O 
*633 

*6i6 
*6oo 
*584 

i.  80  683 

781 

880 

978 

*077 

*I75 

*274 

*372 

*47i 

*569 

1.81  667 
1.82  652 
i.83  638 

1.84  625 
1.85  609 
1.86  595 

1.87  582 
1.88  569 
1.89  556 

766 

75i 
736 

722 
708 
694 

681 
667 
655 

864 
849 
835 
820 
806 
793 

779 
766 

753 

963 
948 

933 
919 

878 
865 
852 

*o6i 
*046 

*032 

*oi8 
*oo4 
990 

977 
964 

95i 

*i6o 
*i45 

*I30 

*u6 

*I02 

*o89 
*075 

*062 

*o5o 

*258 
*244 

*229 

*2I5 
*20I 

*i87 

*I74 
*i6i 
*I48 

*357 
*342 

*328 

*3I3 
*286 

*273 
*260 

*247 

*455 
*44i 

*426 

*4I2 

*398 
*38s 

*37i 
*358 
*346 

*554 
*539 
*525 
*5ii 

*497 
*483 

*47o 
*457 
*445 

i  9Q  543 

1.91  53i 
1.92  519 

1-93  507 
1.94  496 
1.95  485 
1.96  474 

1-97  463 
1.98  452 
1.99442 
2.00  432 

642 

74i 

840 

938 

*037 

*i36 

*235 

*333 

*432 

630 
618 
606 

PI 

573 
562 
55i 
54i 

729 
717 

705 
69^: 
682 
671 
66  1 
650 
640 

827 
815 
804 

770 
760 
749 
739 

926 
914 
903 
891 
880 
869 

859 
848 
838 

*025 
*OI3 
*002 

990 

979 
968 

958 
947 
937 

*I24 
*II2 
*IOO 

*o89 
*078 
*o67 
*o57 
*o46 
*o36 

*223 
*2II 
*I99 

*i88 

*i77 
*i66 

*I56 
*I45 
*i35 

*32I 
*3IO 
*298 

*287 
*276 

*26s 

*254 
*244 

*234 

*420 

*4o8 
*397 
*386 

!3^ 

*364 

*353 
*343 
*333 

53i 

630 

729 

828 

927 

*026 

*I25 

*224 

*323 

B.  0 

1 

2 

8 

4 

5 

0 

7 

8 

9 

Prop.  Pts. 

94 


TABLE  VI. 


log  a  —  log  b  = 
log(«  +  *)  = 

=  A.             log  a  —  log  b  =  B. 
log*  +  (B  -  A).     log  (a  -  b)  =  log  a  -  (B  -  A) 

A. 

B. 

B-A. 

A. 

B. 

B-A. 

A. 

B. 

B-A. 

1.9823 
.9833 
.9842 
.9852 
.9862 

1.9868 
.9878 
.9887 
.9897 
.9907 

.00450 
449 
448 
447 
446 

2.0337 
.0348 

.0359 
.0370 
.0381 

2.0377 
.0388 

•0399 
.0410 
.0421 

.00400 
399 
398 
397 
396 

2.0920 
.0932 
.0945 
.0957 
.0970 

2.0955 
.0967 
.0980 
.0992 
.1005 

.00350  | 

349 
348 
347 
346 

1.9872 
.9882 
.9891 
.9901 
.9911 

1.9917 
.9926 

•9935 
•  9945 
•9955 

.00445 
444 
443 
442 
441 

2.0392 
.0403 
.0414 
.0425 
•  0437 

2.0432 

.0443 
.0454 
.0465 
.0476 

.00395 
394 
393 
392 
39i 

2.0982 
.0995 
.1008 
.1020 
•1033 

2.1017 
.1029 
.1042 
.1054 
.1067 

.00345 
344 
343 
342 
34i 

1.9921 
•9931 
.9941 
.9951 
.9961 

1.9965 
•9975 
.9985 
•9995 
2.0005 

.00440 
439 
438 
437 
436 

2.0448 
.0459 
.0470 
.0481 
.0493 

2.0487 
.0498 
.0509 
.0520 
.0532 

.00390 

389 
388 

387 
386 

2.1046 
.1059 
.1072 
.1085 
.1098 

2.1080 
.1093 
.1106 
.1119 
.1132 

.00340 

339 
338 
337 
336 

1.9971 
.9981 
.9991 

2.0001 
.OOI  I 

2.0015 
.0024 
.0034 
.0044 
.0054 

.00435 
434 
433 
432 
431 

2.0504 
.0515 
.0527 
.0538 
.0550 

2.0543 

•0553 
.0565 
.0576 
.0588 

.00385 
384 
383 
382 
38i 

2.  mi 
.1124 

•  1137 
.1150 
.1163 

2.1144 
.1157 
.1170 
.1183 
.1196 

•00335 
334 
333 
332 
33i 

2.OO2I 
.OO32 
.OO42 
.0052 
.0062 

2.0065 
.0075 
.0085 
.0095 
.0105 

.00430 

429 
428 

427 
426 

2.0561 

.0573 
.0584 
.0596 
.0607 

2.0600 
.0611 
.0622 
.0634 
.0645 

.00380 

379 
378 
377 
376 

2.1176 
.1190 
.1203 
.1216 
.1229 

2.1209 
.1223 
.1236 
.1249 
.1262 

00330 

329 
328 

327 
326 

2.0073 
.0083 
.0093 
.OIO4 
.0114 

2.0115 
.0125 

.0135 
.0146 
.0156 

.00425 
424 
423 
422 
421 

2.0619 
.0630 
.0642 
.0654 
.0666 

2.0656 
.0667 
.0679 
.0691 
.0703 

.00375 
374 
373 
372 
37i 

2.1243 
.1256 
.1270 
.1283 
.1297 

2.1275 
.1288 
.1302 

.1315 
.1329 

.00325 
324 
323 
322 
321 

2.0124 
.0135 
.0145 
.0156 
.0166 

2.0166 
.oi7/ 
.0187 
.0198 
.0208 

.00420 
419 
418 

417 
416 

2.0677 
.0689 
.0701 
.0713 
.0725 

2.0714 
.0726 
.0738 
.0750 
.0762 

.00370 

369 
368 

367 
366 

2.1310 
.1324 
.1338 
.1351 
.1365 

2.1342 
.1356 
.1370 

.1383 
•  1397 

.00320 

319 
3i8 
317 
316  1 

2.0177 
.0187 
.0198 
.0208 
.0219 

2.0218 
.0228 
.0239 
.0249 
.0260 

.00415 
414 

413 
412 
411 

2.0737 
.0749 
.0761 

.0773 
.0785 

2.0773 
.0785 
.0797 
.0809 
.0821 

.00365 
364 
363 
362 
361 

2.1379 
.1393 
.1407 

.1421 
•  H35 

2.1410 

.1424 
.1438 
.1452 
.1466 

.00315 
3'4 

313 
312 

3" 

2.0229 
.0240 
.0251 
.O26l 
.O272 

2.0270 
.0281 
.0292 
.0302 
.0313 

.00410 
409 
408 
407 
406 

2.0797 
.0809 
.0821 
•0833 
.0845 

2.0833 
.0845 
.0857 
.0869 
.0881 

.00360 
359 
358 

352 
356 

2.1449 
.1463 

.1477 
.1491 
.1505 

2.1480 

.1494 
.1508 
.1522 
•  1536 

.00310 

309 
308 
307 
306 

2.O283 
.0294 
.0305 

•OS'S 
.0326 

2.0324 
.0334 
.0345 

•°3II 
.0366 

.00405 
404 

403 
402 
401 

2.0858 
.0870 
.0882 
.0895 
.0907 

2.0893 
.0905 
.0917 
.0930 
.0942 

.00355 
354 
353 
352 
35i 

2.1520 
.1534 
.1548 
.1563 
.'577 

2.1550 
.1564 
.1578 
.1593 
.  1607 

.00305 
304 
303 
302 

30i 

2.0337 

2.0337 

.00400 

2.0920 

2.0955 

.00350 

2.1592 

2  1622 

.00300 

A. 

B. 

B-A. 

A. 

B. 

B-A. 

A. 

B. 

B-A. 

ADDITION  AJND  5UJJTKAUT1U.N  -LUUAK.1T.HM5.            g 

Iog0  —  log£  as  A.             loga  —  log£  =  B. 
log(0  +  V]  =  loga  +  (B  —  A).     log(a  —  V)  as  loga  —  (B  —  A). 

A. 

B. 

B-A. 

A. 

B. 

B-A. 

A. 

B. 

B—  A. 

2.1592 

.1606 

.1621 

.1635 

.1650 

2.1622 

.1636 

.1651 
.1665 
.1680 

.00300 

299 
298 

297 

296 

2.2386 
.2403 
.2421 

.2439 
.2456 

2.2411 
.2428 

.2446 
.2464 
.2481 

.00250 

249 
248 

247 
246 

2.3358 

•3379 
.3401 

.3423 
.3446 

2.3378 
•3399 
.3421 

•  3443 
.3466 

.00200 
199 
198 
197 
196 

2.1665 
.1680 
.1694 

.1710 
.1724 

2.1694 

.1709 
.1723 

.1739 
.1753 

.00295 
294 

293 
292 
291 

2.2474 
.2492 
.2510 
.2528 
.2546 

2.2498 
.2516 

.2534 
.2552 
.2570 

.00245 
244 
243 
242 
241 

2.3468 
.3490 
.3513 
•3535 
.3558 

2.3487 
.3509 
•3532 
•  3554 
•  3577 

.00195 
194 

193 
192 
191 

2.1739 
.1754 
.1770 

.1785 

.1800 

2.1768 
.1783 
.1799 
.1814 
.1829 

.00290 
289 
288 
287 
286 

2.2564 
.2582 
.2600 
.2618 
.2637 

2.2588 
.2606 
.2624 
.2642 
.2661 

.00240 

239 
238 

237 
236 

2.3581 
.3604 
.3627 
.3650 
.3673 

2.3600 

.3646 
.3669 
.3692 

.00190 

187 

1  86 

2.1815 
.1830 
.1846 
.1861 
.1877 

2.1844 
.1858 

.1874 
.1889 
.1905 

to  to  to  to  to 

00  00  0000  OO 
1-4  to  CO  4^>  Cn 

2.2656 
.2674 
.2693 
.2711 
.2730 

2.2679 
.2697 
.2716 
.2734 
•  2753 

.00235 
234 
233 
232 
231 

2.3697 
.3720 

•  3744 
.3768 
.3792 

(0 

CO  CO  CO  CO  OJ 

<->  OO  ONCO  i-i 
O  ON  tO  OOCn 

.00185 
184 

183 
182 

181 

2.1892 
.1908 

.1923 
.1939 
.1955 

2.1920 
.1936 

.1951 
.1967 

.1983 

.00280 
279 

278 

277 
276 

2.2749 
.2768 
.2787 
.2806 

.2825 

2.2772 

.2791 
.2810 
.2829 
.2848 

.00230 
229 
228 
227 
226 

2.3816 
.3840 
.3865 
.3889 
.39H 

.3883 
.3907 
.3932 

.00180 
179 
178 
177 
176 

2.1971 
.1987 

.2002 

.2019 
.2035 

2.1998 
.2014 
.2029 
.2046 
.2062 

.00275 
274 

273 

272 
271 

2.2845 
.2864 
.2884 
.2903 
.2923 

2.2867 
.2886 
.2906 
.2925 
.2945 

.00225 
224 
223 

222 
221 

2-3939 
.3964 
.3989 
.4014 
.4039 

2.3956 
.3981 
.4006 

.4031 
.4056 

.00175 
174 
173 
172 
171 

2  .  205  1 

.2067 

.2083 
.2099 

.2116 

2  .  2078 
.2094 
.2110 
.2126 
.2143 

.00270 

269 
268 
267 
266 

2.2943 
.2962 
.2982 
.3002 
.3022 

2.2965 
.2984 
.3004 
.3024 

.3044 

.OO22O 
219 

218 
217 
216 

2.4065 
.4090 
.4116 
.4142 
.4168 

2.4082 
.4107 
.4133 
.4159 
.4185 

.00170 

\67 
1  66 

2.2132 
.2149 
.2165 
.2182 
.2198 

2.2159 

.2175 
.2191 
.2208 
.2224 

00265 

264 

263 

262 
261 

2.3043 
.3063 
.3083 
.3104 
.3124 

2.3064 
.3084 
.3104 
.3125 
.3H5 

.00215 
214 
213 

212 
211 

2.4195 
.4221 
.4248 

.4275 
.4302 

2.4211 

.4237 
.4264 
.4291 
.4318 

.00165 
164 

'g 

162 

161 

2.2215 

.2232 
.2249 
.2266 
.2283 

2.2241 

.2258 

.2275 

.2292 

.2309 

.00260 

259 
258 

256 

.3166 
.3187 
.3208 
.3229 

2.3166 

.3187 
.3208 

.3229 
.3250 

20g 
208 
207 
206 

2.4329 
.4356 
.4383 
.4411 

•  4439 

2.4345 
.4372 

•4399 
.4427 

•4455 

.00160 

159 
158 

157 

156 

2.2300 

.2317 
.2334 

.2369 

2.2325 
.2342 

.2359 
.2376 

.2394 

.00255 

254 

253 

252 
251 

to 

CO  CO  CO  CO  CO 
CO  CO  tO  tO  tO 
CO  >-i\O  Vjen 
O\4^co  •-•  O 

2.3271 
.3291 
.3313 
•  3334 
.3356 

.OO2O5 
204 
203 
202 
201 

2.4467 
•4495 
.4523 
•4552 
.4581 

2.4482 
.4510 
.4538 
.4567 

.4596 

.00155 
154 
153 
152 

2.2386 

2.2411 

.00250 

2.3358 

2.3378 

.O0200 

2.4609 

2  4624 

.00150 

A. 

B. 

B-A. 

A. 

B. 

B-A. 

A. 

B. 

B-A. 

TABLE  VI. 

1 

Iog0  —  log  b  =  A.             log  a  —  log  b  =  B. 
log  (a  +  £)  =  log*  +  (^  —  ^).     log  (a  —  £)  =  log  a  —  (^  —  ^4). 

A. 

B. 

B-A. 

A. 

B. 

B-A. 

A. 

B. 

B-A. 

2.4609 
.4638 
.4668 
.4697 
.4727 

2.4624 
.4653 
.4683 
.4712 
.4742 

.00150 
149 
148 

H7 
146 

2.6373 
.6416 
.6461 
.6505 
.6550 

2.6383 
.6426 
.6471 

•5Si5 
.6560 

.OOIOO 

.00099 

98 

9£ 
96 

2.9385 
•  9474 
.9563 
.9655 
.9748 

2.9390 
•  9479 
.9568 
.9660 
•9753 

.00050 

J3 
% 

2-4757 
.4787 
.4817 
4848 
4878 

2.4772 
.4801 
.4831 
.4862 
.4892 

.00145 
144 

143 
142 
141 

2.6596 
.6642 
.6688 

.6735 
.6783 

2.6606 
.6651 
.6697 

.6744 
.6792 

.00095 

94 
93 
92 
9i 

2.9844 

2.9941 
3.0041 
.0143 
.0248 

2.9848 
2.9945 
3-0045 
.0147 
.0252 

.00045 
44 
43 
42 

41 

4910 
4941 
4972 
5004 
5036 

2.4924 

•  4955 
.4986 
.5018 
.5050 

.00140 

139 
138 

137 
136 

2.6831 
.6880 
.6928 
.6978 
.7028 

2  .  6840 
.6889 
.6937 
.6987 
.7037 

.00090 
89 
88 

87 
86 

3'°3^ 
.0466 

.0578 
.0694 
.0813 

3-0360 
.0470 
.0582 
.0698 
.0817 

.00040 

P 

! 

5068 
5100 

5J33 
5165 

5199 

2.5081 

.5H| 
.5146 

.5178 
.5212 

.00135 
134 
133 
132 
131 

2.7079 
•  7131 
.7183 
.7236 
.7289 

2.7088 
.7139 
.7191 
.7244 
.7297 

.00085 
84 

83 
82 
81 

3-0935 
.1061 
.1191 
.1324 
.1463 

3-0939 
.1064 
.1194 
.1327 
.1466 

.00035 

34 
33 
32 
3i 

5232 
5266 
5299 

5333 
5368 

2.5245 
.5279 
.5312 
.5346 
.538i 

.00130 
129 
128 
127 
126 

2.7343 
.7398 
•  7453 
.7509 
.7566 

2.7351 
.7406 
.7461 
.7517 

•  7574 

.00080 

79 
78 

8 

3.1606 

.1753 
.1905 
.2063 
.2226 

3.1609 
.1756 
.1908 
.2066 

.2229 

.00030 
29 
28 
27 
26 

5402 
5437 

5472 
5508 
5544 

2.5415 
•  5449 
.5484 
.5520 
.5556 

.00125 
124 
123 

122 
121 

2.7623 
.7682 

.7741 
.7801 
.7862 

2.7631 
.7689 
.7748 
.7808 
.7869 

.00075 
74 
73 
72 
7i 

3-2396 
.2575 
.2760 
.2952 
.3154 

3-2399 
.2577 
.2762 

.2954 
.3156 

.00025 
24 
23 

22 
21 

5580 
5616 

5653 
5690 

5727 

2.5592 
.5628 
.5665 
.5702 

•  5739 

.OOI20 
119 

118 
•117 
116 

2.7923 
.7985 
.8050 
.8114 
.8180 

2.7930 
.7992 
.8057 
.8121 
.8187 

.00070 

69 
68 

67 
66 

3.3366 
•  3590 
.3825 
.4072 
•4335 

3.3368 
•  3592 
.3827 

.4074 
•  4337 

.OO020 
19 

I/ 

16 

2.5765 
5803 
5841 
5880 
.5919 

2.5776 
.5814 
•  5852 
.5891 

.5930 

.00115 
114 
H3 

112 
III 

2.8245 

.8313 
.8381 

.8451 
.8521 

2.8252 
.8319 
.8387 
.8457 
.8527 

.00065 
64 

£3 
62 

61 

3-4617 
•49J7 
.5237 
.5587 
.5964 

3.46i9 
.4918 
.5238 
.5588 

.5965 

.00015 
14 
13 

12 

II 

2.5958 
5998 
.6038 
.6079 
.6120 

2.5969 
.6009 
.6049 
.6090 
.6131 

.OOIIO 

107 
106 

2.8593 
.8666 

.8741 
.8816 
.8893 

2.8599 
.8672 

.8747 
.8822 

.8899 

.00060 
59 
58 

i 

3.6377 
.6835 
•  7345 
.7925 
.8595 

3.6378 
.6836 
.7346 
.7926 
.8596 

.OOOIO 

09 
08 
07 
06 

2.6161 
.6202 
.6244 
.6287 
.6329 

2.6172 
.6212 
.6254 
•6297 
.0339 

.00105 
104 
103 

102 
101 

2.8971 
.9051 
.9132 
.9215 
.9300 

2.8977 
.9056 
•  9'37 
.9220 

.9305 

.00055 
54 
53 
52 
5i 

3.9390 
4.0355 
4.1600 

4.3375 
4.6367 

3.9391 
4.0355 
4.1600 

4.3375 
4.6367 

.00005 

04 
03 

02 

01 

2.6373 

2.6383 

.OOIOO 

2.9385 

2.9390 

.00050 

oo 

00 

.00000 

A. 

B. 

B-A. 

A. 

B. 

B-A. 

A. 

B. 

B-A. 

TABLE  VII.— SQUARES  OF  NUMBERS. 


97 


TABLE  VTL> 

SQUARES  OF  NUMBERS. 

No. 

Square. 

No. 

Square. 

No. 

Square. 

No. 

Square. 

No. 

Square. 

0 

I 
2 

3 

0 

20 

21 
22 
23 

400 

40 

4i 

42 
43 

1600 

60 

61 
62 
63 

3600 

80 

Si 
82 
83 

6400 

I 
4 
9 

441 
484 
529 

1681 
1764 
1849 

3721 
3844 
3969 

6561 
6724 
6889 

4 
5 
6 

16 

25 
36 

24 

25 
26 

576 
625 
676 

44 
45 
46 

1936 
2025 
2116 

64 

65 
66 

4096 
4225 
4356 

84 

85 
86 

7056 
7225 
7396 

7 
8 

9 
10 
ii 

12 
13 

49 
64 
81 

27 
28 
29 

80 

3i 
32 
33 

729 
784 
841 

47 
48 

49 
50 

5i 
52 
53 

2209 
2304 
2401 

67 
68 
69 

70 

7i 
72 

73 

4489 
4624 
4761 

87 
88 
89 

90 

9i 
92 
93 

7569 

7744 
7921 

100 

900 

2500 

4900 

8100 

121 
144 
I69 

961 
1024 
1089 

2601 
2704 
2809 

5041 
5184 
5329 

8281 
8464 
8649 

14 
15 
16 

J96 
225 
256 

34 
35 
36 

1156 

1225 
1296 

54 
55 
56 

2916 
3025 
3136 

74 
75 
76 

5476 
5625 
5776 

94 
95 
96 

8836 
9025 
9216 

17 
18 

19 
20 

289 
324 
36l 

37 
38 
39 

40 

1369 
1444 
7521 

57 

58 
59 

60 

3249 
3364 
348i 

77 
78 
79 

80 

5929 
6084 
6241 

97 
98 
99 

100 

9409 
9604 
9801 

400 

1600 

3600 

6400 

10000 

o8 


SQUARES  OF  NUMBERS  FROM  100  TO  1000. 


1<* 

9~ 

3*+ 

4** 

5~ 

6~ 

1~ 

$•• 

94* 

Diff. 

00 

100 

40O 

900 

1600 

2500 

3600 

4900 

6400 

8100 

00 

X 

01 

102 

404 

906 

1608 

2510 

3612 

4914 

6416 

8118 

01 

O2 
03 

104 
106 

408 
412 

912 
918 

1616 
1624 

252O 
2530 

& 

4928 
4942 

6432 
6448 

8136 
8154 

04 
09 

3 

5 
7 

04 

108 

416 

924 

1632 

2540 

3648 

4956 

6464 

8172 

16 

05 

no 

420 

930 

1640 

2550 

3660 

4970 

6480 

8190 

25 

j  j 

06 

112 

424 

936 

1648 

2560 

3672 

4984 

6496 

8208 

36 

07 

114 

428 

942 

1656 

2570 

3684 

4998 

6512 

8226 

49 

08 

116 

432 

948 

1664 

2580 

3696 

5012 

6528 

8244 

64 

* 

09 

118 

436 

954 

1672 

2590 

3708 

5026 

6544 

8262 

81 

19* 

10 

121 

441 

961 

1681 

2601 

3721 

5041 

6561 

8281 

oo 

21 

II 

123 

445 

967 

1689 

26ll 

3733 

5055 

6577 

8299 

21 

12 

125 

449 

973 

1697 

2621 

3745 

5069 

6593 

8317 

44 

25 

13 

127 

453 

979 

1705 

2631 

3757 

5083 

8335 

69 

27 

14 

129 

457 

985 

1713 

2641 

3769 

5097 

6625 

8353 

96 

20* 

" 

15 

I32 

462 

992 

1722 

2652 

3782 

5112 

6642 

8372 

25 

OT 

16 

134 

466 

998 

1730 

2662 

3794 

5126 

6658 

8390 

56 

J* 

33 

17 

136 

470 

1004 

1738 

2672 

3806 

5140 

6674 

8408 

89 

# 

18 

139 

475 

IOII 

1747 

2683 

3819 

5*55 

6691 

8427 

24 

19 

141 

479 

1017 

1755 

2693 

3831 

5i69 

6707 

8445 

61 

39* 

20 

144 

484 

1024 

1764 

2704 

3844 

5184 

6724 

8464 

00 

4» 

21 

22 

I46 

145 

488 
492 

1030 
1036 

1780 

2714 
2724 

3856 
3868 

5198 
5212 

6740 
6756 

8482 
8500 

84 

43 

AS* 

23 

IS' 

497 

1043 

1789 

2735 

3881 

5227 

6773 

8519 

29 

45 

47 

24 

3 

$ 
I58 

501 
506 

1049 
1056 

IO02 

1797 
1806 
1814 

2745 
2756 
2766 

3893 
3906 

5241 
5256 
5270 

6789 
6806 
6822 

P 
8574 

76 

49* 
5* 
53* 

27 
28 

161 

515 
519 

1069 
1075 

1823 
1831 

2777 
2787 

3931 
3943 

5285 
5299 

6839 
6855 

§?? 

29 
84 

55 

29 

166 

524 

1082 

1840 

2798 

3956 

53H 

6872 

8630 

4i 

59* 

30 

169 

529 

1089 

1849 

2809 

3969 

5329 

6889 

8649 

00 

61 

3i 

171 

533 

1095 

1857 

2819 

3981 

5343 

6905 

8667 

61 

6,* 

32 
33 

:n 

542 

1102 

1108 

1866 
1874 

2830 
2840 

3994 
4006 

5358 
5372 

6922 
6938 

8686 
8704 

n 

"j 
65  | 
67* 

34 

9 

ig 

547 
55J 
556 

1115 

1  122 
1128 

1883 
1892 
1900 

2851 
2862 
2872 

4019 
4032 
4044 

5387 
5402 

6972 
6988 

8723 
8742 
8760 

56 
96 

69* 

7»  / 
73* 

8 

1*0 

566 

"35 
1142 

1909 
1918 

2883 
2894 

4057 
4070 

5431 
5446 

7005 
7022 

8779 

69 

44 

75* 

39 

J93 

57i 

"49 

1927 

2905 

4083 

7039 

88?7 

21 

79* 

40 

196 

576 

1156 

1936 

29l6 

4096 

5476 

7056 

8836 

00 

81 

41 

198 

580 

1162 

1944 

2926 

4108 

5490 

7072 

8854 

81 

8  * 

42 

2OI 

585 

1169 

'953 

2937 

4121 

5505 

7o89 

8873 

64 

8  * 

43 

204 

590 

1176 

1962 

2948 

4134 

5520 

7106 

8892 

49 

87* 

44 

207 

595 

1183 

1971 

2959 

4M7 

5535 

7123 

8911 

36 

89" 

$ 

210 
213 

600 
605 

1190 
1197 

1980 
1989 

2970 
298l 

4160 
4173 

5550 
5565 

7140 
7i57 

8930 
8949 

3 

9t" 
93* 

47 

216 

610 

1204 

1998 

2992 

4186 

558o 

7174 

8968 

09 

OS* 

48 

219 

615 

I2II 

2007 

4199 

5595 

7191 

8987 

04 

95 

07* 

49 

222 

620 

1218 

2016 

3OI4 

4212 

5610 

7208 

9006 

01 

97 
99* 

50 

225 

625 

1225 

2025 

3025 

4225 

5625 

7225 

9025 

oo 

SQUARES  OF  NUMBERS  FROM  100  TO  1000  (Continued). 


99 


1~ 

2~ 

3~ 

4** 

5~ 

6~ 

r«+ 

8~ 

0*^ 

Diff. 

60 

225 

625 

1225 

2025 

3025 

4225 

5625 

7225 

9025 

00 

i 

5i 

228 

630 

1232 

2034 

3036 

4238 

5640 

7242 

9044 

01 

52 
53 

231 
234 

635 
640 

1239 
1246 

2043 
2052 

3047 
3058 

4251 
4264 

5655 
5670 

7259 
7276 

9063 
9082 

04 

09 

3 
5 
7 

54 

237 

645 

1253 

2061 

3069 

4277 

5685 

7293 

9101 

16 

55 

240 

650 

1260 

2070 

3080 

4290 

5700 

7310 

9120 

25 

56 

243 

655 

1267 

2079 

3091 

4303 

5715 

7327 

9U9 

36 

131  ' 

57 

246 

660 

1274 

2088 

3102 

43l6 

5730 

7344 

9158 

49 

58 

249 

665 

1281 

2097 

31*3 

4329 

5745 

9177 

64 

59 

252 

670 

1288 

2106 

3124 

4342 

5760 

7378 

9196 

Si 

19* 

60 

256 

676 

1296 

2116 

3136 

4356 

5776 

7396 

9216 

00 

•I 

61 

259 

681 

1303 

2125 

3U7 

4369 

5791 

7413 

9235 

21 

62 
63 

265 

686 
691 

1310 
1317 

2134 
2143 

3158 
3169 

4382 
4395 

5806 
5821 

7430 
7447 

9254 
9273 

% 

35 
27 

64 

268 

696 

1324 

2152 

3l80 

4408 

5836 

7464 

9292 

96 

29* 

272 

702 

1332 

2162 

3192 

4422 

5852 

7482 

9312 

25 

66 

275 

707 

'339 

2171 

3203 

4435 

5867 

7499 

9331 

56 

33 

% 

278 

712 

718 

1346 
1354 

2180 
2190 

32H 
3226 

4448 
4462 

5882 
5898 

7534 

9350 
9370 

89 
24 

»* 

69 

285 

723 

1361 

2199 

3237 

4475 

59^3 

755' 

9389 

61 

39* 

70 

289 

729 

1369 

2209 

3249 

4489 

5929 

7569 

9409 

00 

41 

72 

292 
295 

734 
739 

1376 
1383 

2218 
2227 

3260 
3271 

4502 
4515 

5944 
5959 

7586 
7603 

9428 
9447 

41 

84 

43 

73 

299 

745 

I39i 

2237 

3283 

4529 

5975 

7621 

9467 

29 

47 

74 

9 

302 
306 
309 

750 

1398 
1406 

1413 

2246 
2256 
.2265 

3294 
3306 

4542 
4556 
4569 

5990 
6006 
6021 

7638 

7656 

7673 

9486 
9506 
9525 

76 

49* 

s- 

78 

313 
316 

767 

772 

1421 
1428 

2275 
2284 

3329 
3340 

4583 
4596 

6037 
6052 

7691 
7708 

9545 
9564 

29 

84 

55 

79 

320 

778 

1436 

2294 

3352 

4610 

6068 

7726 

9584 

41 

57* 
59* 

80 

324 

784 

1444 

2304 

3364 

4624 

6084 

7744 

9604 

00 

61 

81 

327 

789 

I45I 

2313 

3375 

4637 

6099 

7761 

9623 

61 

fia* 

82 

331 

795 

U59 

2323 

3387 

4651 

6115 

7779 

9643 

24 

03 

6. 

83 

334 

800 

1466 

2332 

3398 

4664 

6130 

7796 

9662 

89 

J 

67* 

f4 

ii 

338 
342 
345 

806 
812 
817 

1474 
1482 
1489 

2342 
2352 
2361 

3422 
3433 

4678 
4692 
4705 

6146 
6162 
6177 

78H 
7832 
7849 

9682 
9702 
9721 

56 

69* 
71 

73* 

87 

349 

823 

1497 

2371 

3445 

4719 

6193 

7867 

9741 

69 

88 

353 

829 

1505 

2381 

3457 

4733 

6209 

7885 

9761 

44 

75 

89 

357 

835 

'513 

2391 

3469 

4747 

6225 

79°3 

9781 

21 

77* 
79* 

90 

361 

841 

1521 

2401 

348i 

4761 

6241 

7921 

9801 

00 

81 

91 

364 

846 

1528 

2410 

3492 

4774 

6256 

793« 

9820 

81 

92 
93 

368 
372 

852 
858 

1536 
1544 

2420 
2430 

3504 
3516 

4788 
4802 

6272 
6288 

7956 

^Q74 

9840 
0860 

64 
49 

83* 
85' 
87* 

94 

384 

864 
870 
876 

1560 
1568 

2440 
2450 
2460 

3528 
3540 
3552 

4816 

4830 
4844 

6304 
6320 
6336 

79^* 
8010 
8028 

9880 
9900 
9920 

36 

s 

89* 
9** 
93* 

11 

388 
392 

882 
888 

1576 

2470 
2480 

3564 
3576 

4858 
4872 

6352 
636* 

8046 
8064 

9940 
9960 

09 
04 

95* 

99 

396 

894 

1592 

2490 

4886 

6384 

$082 

9980 

01 

97* 
99* 

100 

400 

900 

1600 

2500 

3600 

4900 

6400 

8100 

1  0000 

oo 

TABLE  VIII.— DECIMALS  OF  DAY  INTO  HOURS,  ETC. 


101 


H.  M.  S. 

H.M.S. 

H.  M.  S. 

H.M.S. 

D. 

H.  M.  S. 

~"~      o 

D. 

H.  M.  S. 

IOO 

IOO 

IOO 

IOO* 

d. 

h.  m.  s. 

M.   S. 

s. 

d. 

h.  m.  s. 

m,  s. 

*. 

O.OI 

o  14  24 

o  8.64 

0.09 

0.51 

12  14  24 

7  20.64 

4.41 

0.02 

o  28  48 

o  17.28 

0.17 

0.52 

12  28  48 

7  29.28 

4.49 

0.03 

o  43  12 

o  25.92 

0.26 

0.53 

12  43  12 

7  37.92 

4.58 

O.O4 

o  57  36 

o  34.56 

0.35 

0-54 

12  57  36 

7  46.56 

4.67 

O.O5 

I  12  0 

o  43.20 

o.43 

0.55 

13  12  0 

7  55-20 

4.75 

O.O6 

I  26  24 

o  51.84 

0.52 

0.56 

13  26  24 

8  3.84 

4.84 

0.07 

I  40  48 

0.48 

0.60 

0.57 

13  40  48 

8  12.48 

4.92 

0.08 

I  55  12 

9.12 

0.69 

0.53 

13  55  12 

8  21.12 

5.01 

O.O9 
0.10 

2  9  36 

2  24  0 

17.76 
26.40 

0.78 
0.86 

0-59 
0.60 

14  9  36 
14  24  o 

8  29.76 
8  38.40 

5.10 
5.18 

O.II 

2  38  24 

35.04 

0.95 

0.61 

14  38  24 

8  47.04 

5.27 

0.12 

2  52  48 

43-68 

.04 

0.62 

14  52  48 

8  55.68 

5.36 

0.13 

3  7  12 

52.32 

.12 

0.63 

15  7  12 

9  4.32 

5.44 

0.14 
0.15 

3  21  36 
3  36  o 

2  0.96 
2  9.60 

.21 
.30 

0.64 
0.65 

15  21  36 

15  36  o 

9  12.96 
9  21.60 

III 

0.16 
0.17 

3  50  24 
4  4  48 

2  18.24 
2  26.88 

.38 

•  47 

0.66 
0.67 

15  50  24 

16  448 

9  30-24 
9  38.88 

5.70 
5.79 

0.18 

4  19  12 

2  35.52 

.56 

0.68 

16  19  12 

9  47.52 

5.88 

0.19 

4  33  36 

2  44.16 

.64 

0.69 

16  33  36 

9  56.16 

5.96 

O.2O 

448  o 

2  52.80 

•73 

0.70 

16  48  o 

10  4.80 

6.05 

0.21 

5  2  24 

3  L44 

.81 

0.71 

17  2  24 

10  13.44 

6.13 

0.22 

5  16  48 

3  10.08 

.90 

0.72 

17  16  48 

10  22.08 

6.22 

0.23 

5  31  I2 

3  18.72 

•99 

0.73 

17  31  12 

10  30.72 

6-31 

0.2l 
0.2$ 

5  45  36 
600 

3  27.36 
3  36.00 

2.07 
2.16 

0.74 
o.75 

17  45  36 
1800 

10  39.36 
10  48.00 

6.39 
6.48 

O.26 

6  14  24 

3  44.64 

2.25 

0.76 

18  14  24 

10  56.64 

6-57 

0.27 

6  28  48 

3  53.28 

2-33 

o.77 

18  28  48 

II  5.28 

6.65 

0.28 

6  43  12 

4  1.92 

2.42 

0.78 

18  43  12 

II  13.92 

-  6.74 

0.29 

6  57  36 

4  10.56 

2.51 

0.79 

18  57  36 

II  22.56 

6.83 

0.30 

7  12  0 

4  19.20 

2.59 

0.80 

19  12   O 

II  31.20 

6.91 

0.31 

7  26  24 

4  27.84 

2.68 

0.81 

19  26  24 

II  39.84 

7.00 

0.32 

7  40  48 

4  36.48 

2.76 

0.82 

19  4O  48 

II  48.48 

7.08 

0-33 

7  55  12 

4  45.12 

2.85 

0.83 

19  55  12 

II  57.12 

7-17 

8  9  36 

2.94 

0.84 

20  9  36 

12   5.76 

7.26 

0.35 

8  24  o 

5  2.40 

3.02 

0.85 

20  24  o 

12  14.40 

7.34 

0.36 

8  38  24 

5  ".04 

3  TI 

0.86 

20  38  24 

12  23.04 

7-43 

0.37 

8  52  48 

5  19.68 

3-20 

0.87 

20  52  48 

12  31.68 

7.52 

0.38 

9  7  12 

5  28.32 

3.28 

0.88 

21   7  12 

12  40.32 

7.6o 

o-39 

9  21  36 

5  36.96 

3-37 

0.89 

21  21  36 

12  48.96 

7.69 

0.40 

9  36  o 

5  45-60 

3-46 

0.90 

21  36   0 

12  57.6O 

7.78 

0.41 

9  50  24 

5  54-24 

3-54 

0.91 

21  50  24 

13  6.?A 

7.86 

0.42 

10  4  48 

6  2.88 

3.63 

0.92 

22  4  48 

13  14.88 

7.95 

o.43 

IO  19  12 

6  11.52 

3-72 

0.93 

22  19  12 

I?  13.52 

8.04 

0.44 

10  33  36 

6  20.16 

0.94 

22  33  36 

15,  jz.ib 

8.12 

0.45 

10  48  o 

6  28.80 

3-89 

0.95 

22  48  0 

15  40.80 

8.21 

0.46 
0.47 

II   2  24 

ii  16  48 

6  37-44 
6  46.08 

3-97 
4.06 

0.96 
o.97 

23  2  24 

23  16  48 

13  49-44 
13  58.08 

8.29 
8.38 

0.48 

II  31  12 

6  54.72 

4.15 

0.98 

23  31  12 

14  6.72 

8.47 

0.49 

ii  45  36 

7  3.36 

4.23 

0-99 

23  45  36 

14  15.36 

8.55 

0.50 

12  0  0 

7  12.00 

4-32 

1.  00 

24  o  o 

14  24.00 

8.64 

i 

102 


TABLE  IX.— ARC  INTO  TIME  AND  VICE  VERSA. 


o 

h,  m. 

o 

/i.  m. 

0 

h.  m. 

0 

h.  m. 

o 

h.  m. 

0 

h.  m. 

i 

m.  s. 

„ 

j. 

o 

O  O 

60 

4  o 

1  20 

8  0 

180 

12  O 

240 

16  o 

300 

20  0 

0 

0  0 

o 

0.000 

I 

o  4 

61 

4  4 

121 

8  4 

181 

12  4 

241 

16  4 

301 

20  4 

i 

o  4 

i 

0.066 

2 

o  8 

62 

4  8 

122 

8  8 

182 

12  8 

242 

16  8 

302 

20  8 

2 

o  8 

2 

0  *33 

3 

0  12 

63 

4  12 

123 

8  12 

183 

12  12 

243 

16  12 

3°3 

2O  12 

3 

0  12 

3 

0.200 

4 

o  16 

64 

4  16 

I24 

8  16 

184 

12  16 

244 

16  16 

304 

20  16 

4 

o  16 

4 

0.266 

I 

o  20 
o  24 

66 

420 

I25 
126 

8  20 

824 

185 
186 

12  20 
12  24 

III 

16  20 
16  24 

305 
306 

20  20 

20  24 

I 

O  20 

o  24 

\ 

o  333 
0.400  i 

7 

028 

67 

4  28 

127 

8  28 

187 

12  28 

247 

16  28 

20  28 

7 

0  28 

7 

0.466 

8 

032 

68 

4  32 

128 

832 

188 

12  32 

248 

16  32 

308 

2o  32 

8 

o  32 

8 

o-533 

9 

o  36 

69 

436 

129 

836 

189 

12  36 

249 

1636 

309 

20  36 

9 

o  36 

9 

0.600 

10 

o  40 

70 

4  40 

130 

8  40 

190 

»  40 

250 

16  40 

310 

2O  40 

IO 

o  40 

10 

0.666 

n 

o  44 

4  44 

844 

191 

12  44 

251 

16  44 

3" 

20  44 

n 

o  44 

n 

0-733 

12 

o  48 

72 

448 

132 

8  48 

192 

12  48 

252 

1648 

312 

2048 

12 

o  48 

12 

0.800 

13 

052 

73 

452 

133 

852 

193 

12  52 

253 

16  52 

313 

20  52 

«3 

o  52 

13 

0.866  i 

'4 

o  56 

74 

456 

134 

8  56 

194 

12  56 

254 

16  56 

20  56 

M 

o  56 

M 

0-933 

11 

o 
4 

76 

c  o 
5  4 

135 
136 

Q  O 

9  4 

'95 
196 

13  o 
13  4 

255 
256 

17  o 
17  4 

3'5 
316 

21  O 
21  4 

15 

16 

o 

4 

15 

1  6 

1.  000 

1.066 

11 

8 

12 

78 

5  8 

5  '2 

137 
138 

9  8 
9  12 

197 
198 

13  8 

13  12 

257 
258 

17  8 
17  12 

317 

21   8 
21  12 

\l 

8 

12 

M 

I  133 

1.200 

*9 

16 

79 

516 

139 

9  16 

199 

13  16 

259 

17  16 

319 

21  16 

19 

16 

19 

1.266 

20 

20 

80 

5  20 

140 

9  20 

200 

13  20 

260 

17  20 

320 

21  20 

20 

20 

20 

'•333 

21 
22 

24 
28 

Si 
82 

528 

141 

142 

9  24 
9  28 

201 

202 

13  24 

n  28 

261 
262 

17  24 
17  28 

321 
322 

21  24 
21  28 

21 
22 

3 

21 

22 

i  400 
1.466 

23 

32 

83 

5  3? 

143 

9  32 

203 

'3  32 

263 

17  32 

323 

21  32 

23 

32 

23 

'  533 

24 

36 

84 

536 

144 

9  36 

204 

13  36 

264 

17  36 

324 

21  36 

24 

36 

24 

i.  600 

11 

40 
44 

II 

5  40 
5  44 

MS 

146 

9  40 
9  44 

205 
206 

«3  40 

»3  44 

266 

1740 

'7  44 

1 

21  40 

21  44 

3 

40 
44 

11 

1.666 
1-733 

11 

48 
52 

11 

548 
5  52 

H8 

948 
9  52 

207 
208 

13  48 
13  52 

267 
268 

1748 
17  52 

21  48 
21  52 

11 

48 
52 

11 

i.  806 
1.866 

29 

56 

89 

5  56 

149 

9  S^ 

209 

*3  56 

269 

17  56 

329 

21  56 

29 

56 

29 

*  933 

30 

2  O 

90 

6  o 

ISO 

1C  O 

210 

14  o 

270 

18  o 

330 

22  O 

30 

2  O 

30 

2  OOO 

31 

2  4 

6  3 

10  4 

211 

14  4 

271 

18  4 

331 

22  4 

2  4 

31 

2.066 

32 

2  8 

92 

6  8 

152 

ic  8 

212 

14  8 

272 

18  8 

332 

22  8 

32 

2  8 

32 

2-133 

33 

2  12 

93 

6  12 

153 

10  12 

213 

14  12 

273 

18  12 

333 

22  12 

33 

2  12 

33 

2.200 

34 

2  16 

94 

6  16 

154 

ic  16 

2I4 

14  16 

274 

18  16 

334 

22  16 

34 

2  16 

34 

2.266 

35 
36 

2  20 
2  24 

95 

96 

6  20 

624 

155 

IO  20 

ic  24 

215 

216 

14  20 
14  24 

III 

18  20 
18  24 

335 
336 

122  20 
22  24 

35 
36 

2  20 
2  24 

1 

2-333 
2.4OO 

37 

2  28 

6  28 

*57 

10  28 

217 

14  28 

277 

18  28 

337 

22  28 

2  28 

2.466 

38 

232 

98 

632 

158 

10  32 

218 

M  32 

278 

1832 

338 

22  32 

38 

2  32 

38 

2-533 

39 

236 

99 

6  36 

159 

10  36 

219 

M  36 

279 

18  36 

339 

22  36 

39 

2  36 

39 

2.600 

40 

240 

IOO 

640 

160 

10  40 

22O 

M  40 

280 

18  40 

340 

22  40 

40 

2  40 

40 

2.666 

4' 

2  44 

101 

644 

161 

10  44 

221 

M  44 

281 

18  44 

22  44 

2  44 

2  733 

4? 

248 

102 

6  48 

162 

10  48 

222 

1448 

282 

18  48 

342 

22  48 

42 

248 

42 

2.800 

43 

2  52 

103 

652 

163 

10  52 

223 

14  52 

283 

18  52 

343 

22  S2 

43 

2  52 

43 

2.866 

44 

2  56 

104 

r  «;6 

164 

10  56 

224 

14  56 

284 

18  56 

344 

2256 

44 

256 

44 

2  933 

45 

46 

3  o 
3  4 

I05 
1  06 

7  4 

£ 

II   0 

ii  4 

III 

15  o 
15  4 

III 

19  o 

19  4 

345 
346 

23  o 

23  4 

46 

3  o 
3  4 

46 

3.000 
3.066 

47 

3  8 

107 

7  8 

167 

n  8 

227 

15  8 

287 

19  8 

347 

23  8 

47 

3  8 

47 

3  133 

48 

3  I2 

108 

7  12 

1  68 

II  12 

228 

15  12 

288 

19  12 

348 

23  12 

48 

3  I2 

4^ 

3.200 

49 

316 

109 

716 

169 

ii  16 

229 

15  16 

289 

19  16 

349 

23  16 

49 

316 

49 

3.266 

50 

320 

no 

7  20 

170 

II  20 

230 

15  20 

290 

19  20 

350 

23  20 

50 

3  20 

50 

3-333 

51 

3  24 

in 

724 

171 

II  24 

23I 

15  24 

291 

19  24 

351 

23  24 

51 

51 

3.400 

52 

3  28 

112 

728 

172 

II  28 

232 

15  28 

292 

19  28 

352 

23  28 

52 

3  28 

52 

3.466 

53 

3  32 

"3 

732 

173 

II  32 

233 

15  32 

293 

19  32 

353 

23  32 

53 

3  32 

53 

3  533 

54 

336 

114 

736 

174 

II  36 

234 

15  36 

294 

19  36 

354 

2336 

54 

336 

54 

3-600 

55 

3  40 

115 

740 

175 

II  40 

235 

'5  40 

295 

19  40 

355 

23  40 

55 

3  40 

55 

3.666 

56 

3  44 

no 

7  44 

176 

ii  44 

236 

15  44 

296 

19  44 

356 

23  44 

56 

3  44 

56 

3  733 

348 

117 

748 

II  48 

237 

15  48 

297 

1948 

357 

2348 

348 

57 

3.800 

58 

3  52 

118 

7  52 

178 

II  52 

238 

15  52 

298 

19  52 

358 

23  52 

58 

3  52 

58 

3-866 

.59 

356 

119 

756 

179 

II  56 

239 

15  56 

299 

19  56 

359 

2356 

59 

356 

59 

3  933 

TABLE  Xa.— TO  CONVERT  MEAN  INTO  SIDEREAL  TIME. 


103 


Mean  T. 
h.    m. 

Correction. 
m.       s. 

Mean  T. 
h.     m. 

Correction. 
/«.     s. 

Mean  T 
h.    m. 

Correction. 
m.      s. 

Corr.  for  min. 
and  sec. 
m.  s.       s. 

o      o 

0        0.00 

8      o 

i     18.85 

16      o 

2      37.70 

0     10 

0.03 

10 

1.64 

10 

20.50 

10 

39.35 

20 

O.Oj 

20 

3.29 

20 

22.14 

20 

40.99 

30 

0.08 

30 

4-93 

30 

23.78 

30 

42.63 

40 

O.II 

40 

6.57 

40 

25-42 

40 

44.28 

5° 

0.14 

50 

8.21 

50 

27.07 

50 

45.92 

I        0 

0.16 

IO 

0.19 

I         0 

o     9.86 

9     o 

i    28.71 

17     o 

2     47.56 

20 
3° 

0.22 
C.25 

IO 

11.50 

IO 

30.35 

IO 

49.20 

40 

0.27 

20 

13.14 

20 

31-99 

20 

50.85 

0.30 

30 

14.78 

30 

33.64 

30 

52.49 

2       O 

0.33 

40 
50 

16.43 
16.07 

40 
50 

35.28 
36.92 

40 
50 

54.13 
55-77 

XO 
20 

0.36 
0.33 

30 

0.41 

2       0 
10 

o    19.71 
21.36 

10       0 

10 

i    38.56 
40.21 

18      o 

10 

2      57.42 
59.06 

40 
50 

0.44 
0.47 

20 

23.00 

20 

41.85 

20 

3      0.70 

3     o 

o  49 

30 

c    40 

So 

24.64 
26.28 
27.93 

30 
40 
50 

43-49 
45.14 
46.78 

30 
40 
50 

2.34 

3.99 
5.63 

IO 
20 

30 
40 

0.52 
o  55 

o  60 

50 

0.63 

3     o 

o    29.57 

II        0 

i    48.42 

19      o 

3      7.27 

4      ° 

0.66 

10 

31.21 

10 

50.06 

10 

8.92 

10 

0.68 

20 

32.86 

20 

51.71 

20 

10.56 

20 

0.71 

30 

34.50 

3° 

53-35 

30 

12.  2O 

30 

0.74 

40 

36.14 

40 

54-99 

40 

13.84 

40 

0.77 

50 

37.78 

56.64 

50 

15.49 

5° 

0.79 

c      o 

0.82 

4     o 

o    39-43 

12        0 

I     58.28 

20     o 

3    I7.I3 

IO 

o  85 

n  RS 

10 

20 
30 

41.07 
42.71 

44.35 

10 
20 
30 

59-92 
2        1.56 

10 
20 
30 

18.77 
20.42 
22.06 

20 

30 

40 
5° 

o.oo 

0.90 

093 
0.96 

40 

46.00 

40 

4*85 

40 

23.70 

50 

47.64 

50 

6.49 

50 

25.34 

6     o 

10 

o  99 

.01 

20 

04 

5     o 

o   49.28 

13     o 

2       8.13 

21        0 

3    26.99 

30 

.07 

10 

50.92 

10 

9.78 

10 

28.63 

40 

IO 

20 

52  57 

20 

11.42 

20 

30.27 

50 

.12 

30 

54.21 

30 

13.06 

30 

31.91 

7      O 

j- 

40 

55.85 

40 

14.70 

40 

33.56 

/ 

10 

'18 

50 

57.50 

50 

16.35 

50 

35-20 

20 

.21 

30 

23 

6     o 

10 

o    59.H 
i      0.78 

14     o 

10 

2      17.99 
19.63 

22       0 
10 

3    36.84 
38.48 

40 
50 

.29 

20 

2.42 

20 

21.28 

20 

40.13 

8     o 

.31 

30 

4.07 

30 

22.92 

30 

4L77 

IO 

34 

40 

5.71 

40 

24.56 

40 

43-41 

20 

37 

50 

7-35 

50 

26.20 

50 

45.o6 

3«> 

40 

.40 
.42 

So 

45 

7     o 

10 

I      9.00 

10.64 

15     o 

10 

2     27.85 
29.49 

23      o 

10 

3    46.70 
48.34 

9     o 

48 

rf\ 

20 

12.28 

20 

20 

49.98 

20 

5° 

30 

13.92 

30 

32.77 

30 

51-63 

3° 

•  ?o 

40 

15.57 

40 

34.42 

40 

53.27 

40 

Pg 

50 

17.21 

50 

36.06 

50 

50 

.6, 

TABLE  X£.— TO  CONVERT  SIDEREAL  INTO  MEAN  TIME. 


Sid.  T. 
k.    m. 

Correction. 
m.       s. 

Sid.  T. 
A.     m. 

Correction. 
m.      s. 

Sid.  T. 
k.    m. 

Correction, 
w.      s. 

Corr.  for  min. 
and  sec. 
m.  s.       s. 

0        0 

o      o.oo 

8      o 

I     18.64 

16      o 

2      37-27 

O      IO 

0.03 

10 
20 

1.64 
3.28 

10 

20 

20.28 
21  .91 

IO 

20 

38.91 
40.55 

20 

30 

0.05 
0.08 

30 

4.92 

30 

23.55 

30 

42.19 

40 

O.1I 

40 

6-55 

40 

25.19 

40 

43.83 

5° 

0.14 

50 

8  19 

50 

26.83 

50 

45.46 

I         0 

o.  16 

10 

o.  19 

t        0 

o     9.83 

9      o 

I     28.47 

17      o 

2    47.10 

20 

3° 

0.22 
0.25 

IO 

11.47 

IO 

30.10 

10 

48.74 

40 

O.27 

20 

13." 

20 

3L74 

20 

50.38 

50 

0.30 

30 

H.74 

30 

33.38 

30 

52.02 

2        O 

o  •?  3 

40 

16.38 

40 

35-02 

40 

53-66 

10 

0.35 

50 

18.02 

50 

36.66 

50 

55-29 

20 

0.38 

30 

0.41 

t        0 

IO 

o    19.66 
21.30 

10        0 
10 

i     38.30 
39-93 

18      o 

IO 

2      56.93 
58.57 

40 
50 

0.44 
0.47 

20 

22.94 

20 

4L57 

20 

3      0.21 

3        O 

0.49 

30 

40 

24.57 
26.21 

30 

40 

43-21 
44-85 

30 
40 

1.85 

3.48 

IO 
90 

0.52 

o-55 

50 

27.85 

50 

46.49 

50 

5.12 

3° 
40 

0.57 
0.60 

50 

0.63 

^      o 

o    29.49 

II        0 

i    48.12 

19     o 

3      6.76 

4-      o 

0.66 

10 

31.13 

10 

49.76 

10 

8.40 

10 

0.68 

20 

32.76 

20 

51.40 

20 

10.04 

20 

0.71 

30 

34.40 

30 

53.04 

30 

11.68 

30 

0.74 

40 

36.04 

40 

54.68 

40 

13.32 

40 

0.76 

50 

37.68 

50 

56.32 

50 

J4-95 

5° 

0.79 

5     ® 

0.82 

4      o 

10 
20 

o    39.32 
40.96 
42.60 

12        0 
IO 
2O 

i     57.96 

59-59 
2      1.23 

20      o 

10 

20 

3     16.59 
18.23 
19.87 

10 
20 
30 

4° 

0.85 
0.87 
0.90 
0.93 

30 

44.23 

30 

2.87 

30 

21.51 

5° 

0.96 

40 

45.87 

40 

4-5' 

40 

23.14 

30 

47.51 

50 

6.15 

50 

24.78 

6      o 

10 

0.98 
i  .01 

20 

i  .04 

C         o 

o    49.15 

13      o 

2        7.78 

21        0 

3    26.42 

30 

i.  06 

10 

50.79 

IO 

9.42 

IO 

28.06 

40 

1.09 

20 

52  42 

20 

II.  06 

20 

29.70 

5° 

I  .  12 

30 

54.06 

30 

12.70 

30 

31.34 

7     ° 

.15 

40 

55-70 

40 

14-34 

40 

32.97 

IO 

•  17 

50 

57-34 

50 

15.98 

50 

34.6i 

20 

.20 

30 

23 

6      o 

o    58.98 

14      o 

2      I7.6l 

22        0 

3    36.25 

40 

fr\ 

.26 

10 

I      0.62 

10 

19.25 

10 

37.89 

5° 

l2 

20 

2.25 

20 

20.89 

20 

39-53 

8      o 

•31 

-30 

3.89 

30 

22.53 

30 

41.16 

10 

•34 

40 

5-53 

40 

24.17 

40 

42.80 

20 

•37 

50 

7.17 

50 

25.80 

50 

44-44 

3° 
40 

•39 
.42 

5° 

•45 

7      o 

i      8.81 

15      o 

2      27.44 

23      o 

3    46.08 

4*7 

IO 

10.44 

lo 

29.08 

10 

47.72 

IO 

•*t/ 

20 

12.08 

20 

30.72 

20 

49.36 

20 

C? 

30 

13.72 

30 

32.36 

30 

51.00 

3° 

•56 

40 

15.36 

40 

34-00 

40 

52.63 

40 

.58 

5° 

17.00 

50 

35.64 

50 

54.27 

50 

.61 

i 

UNIVERSITY  OF  CALIFORNIA  LIBRARY 
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